The document discusses different types of equations that can represent straight lines in a plane, including point-slope form, two-point form, slope-intercept form, and normal form. It provides examples of writing the equation of a line given characteristics like two points, slope and intercept, or being parallel/perpendicular to another line. The document also covers topics like finding the distance from a point to a line and the equations of angle bisectors.
The document discusses sets of axioms and finite geometries. It begins by defining geometry as "earth measure" from its Greek roots and discusses some early examples of geometry like Eratosthenes' measurement of the circumference of the Earth. It then discusses the development of geometry through the Greeks and Euclid, including undefined terms, axioms, postulates, and theorems. It provides examples of Euclid's axioms and postulates and discusses modern refinements. Finally, it introduces finite geometries, which have a finite number of elements, and provides an example of a three-point geometry with its axioms and a proof of one of its theorems.
This document provides a module on linear functions. It defines linear functions as those that can be written in the form f(x) = mx + b, where m is the slope and b is the y-intercept. The module teaches how to determine if a function is linear, rewrite linear equations in slope-intercept form, and graph linear functions given various inputs like two points, x- and y-intercepts, slope and a point, or slope and y-intercept. Examples and practice problems are provided to help students learn to identify, write, and graph different types of linear functions.
Quadrilateral that are parallelogram.pptxRizaCatli2
The document discusses quadrilaterals and parallelograms. It provides examples of different types of quadrilaterals like trapezoids, parallelograms, rhombuses, squares, and kites. It asks students to identify quadrilaterals and parallelograms in different shapes. It also discusses the properties of parallelograms and provides activities for students to practice identifying and classifying quadrilaterals.
This document discusses key concepts about circles, including:
- The standard equation of a circle is (x - h)2 + (y - k)2 = r2, where (h, k) are the coordinates of the center and r is the radius.
- Given the equation or properties of a circle, one can determine its center and radius or write the equation in standard form.
- Points can lie inside, outside, or on a circle, which can be determined by comparing distances or substituting into the equation.
- A circle and line can intersect in 0, 1, or 2 points, which can be found using algebraic techniques.
- The equation of a circle can be found given 3
This document introduces the distance formula, which is used to calculate the distance between two points (x1, y1) and (x2, y2) on a coordinate plane. The distance formula is the square root of (x1 - x2) squared plus (y1 - y2) squared. Several examples are worked through to demonstrate finding the distance between points using their coordinates. Practice problems are also provided for the reader to work through on their own.
This document provides information about hyperbolas including their definition as the locus of points where the difference between the distances to two fixed points (foci) is constant. It gives the standard equation forms for horizontal and vertical hyperbolas and discusses their key properties like asymptotes. Examples of using inequalities to determine if points lie inside or outside the region bounded by a hyperbola are provided. The document also includes exam questions testing the understanding of hyperbolas and their equations.
The document discusses different types of equations that can represent straight lines in a plane, including point-slope form, two-point form, slope-intercept form, and normal form. It provides examples of writing the equation of a line given characteristics like two points, slope and intercept, or being parallel/perpendicular to another line. The document also covers topics like finding the distance from a point to a line and the equations of angle bisectors.
The document discusses sets of axioms and finite geometries. It begins by defining geometry as "earth measure" from its Greek roots and discusses some early examples of geometry like Eratosthenes' measurement of the circumference of the Earth. It then discusses the development of geometry through the Greeks and Euclid, including undefined terms, axioms, postulates, and theorems. It provides examples of Euclid's axioms and postulates and discusses modern refinements. Finally, it introduces finite geometries, which have a finite number of elements, and provides an example of a three-point geometry with its axioms and a proof of one of its theorems.
This document provides a module on linear functions. It defines linear functions as those that can be written in the form f(x) = mx + b, where m is the slope and b is the y-intercept. The module teaches how to determine if a function is linear, rewrite linear equations in slope-intercept form, and graph linear functions given various inputs like two points, x- and y-intercepts, slope and a point, or slope and y-intercept. Examples and practice problems are provided to help students learn to identify, write, and graph different types of linear functions.
Quadrilateral that are parallelogram.pptxRizaCatli2
The document discusses quadrilaterals and parallelograms. It provides examples of different types of quadrilaterals like trapezoids, parallelograms, rhombuses, squares, and kites. It asks students to identify quadrilaterals and parallelograms in different shapes. It also discusses the properties of parallelograms and provides activities for students to practice identifying and classifying quadrilaterals.
This document discusses key concepts about circles, including:
- The standard equation of a circle is (x - h)2 + (y - k)2 = r2, where (h, k) are the coordinates of the center and r is the radius.
- Given the equation or properties of a circle, one can determine its center and radius or write the equation in standard form.
- Points can lie inside, outside, or on a circle, which can be determined by comparing distances or substituting into the equation.
- A circle and line can intersect in 0, 1, or 2 points, which can be found using algebraic techniques.
- The equation of a circle can be found given 3
This document introduces the distance formula, which is used to calculate the distance between two points (x1, y1) and (x2, y2) on a coordinate plane. The distance formula is the square root of (x1 - x2) squared plus (y1 - y2) squared. Several examples are worked through to demonstrate finding the distance between points using their coordinates. Practice problems are also provided for the reader to work through on their own.
This document provides information about hyperbolas including their definition as the locus of points where the difference between the distances to two fixed points (foci) is constant. It gives the standard equation forms for horizontal and vertical hyperbolas and discusses their key properties like asymptotes. Examples of using inequalities to determine if points lie inside or outside the region bounded by a hyperbola are provided. The document also includes exam questions testing the understanding of hyperbolas and their equations.
Problem solving involving polynomial functionMartinGeraldine
The document provides examples of solving geometry problems involving finding dimensions of objects given certain constraints. The first example involves finding the size of a wood sheet needed to construct a wooden tray given its volume. The second example involves finding the lengths of the legs of a right triangle given its area and the relationship between the legs. Both examples involve setting up equations based on the given information and solving using algebraic steps to find the desired dimensions.
Analytic geometry introduced in the 1630s by Descartes and Fermat uses algebraic equations to describe geometric figures on a coordinate system. It connects algebra and geometry by plotting points using a coordinate system with real number coordinates. This allows geometric shapes to be represented by algebraic equations which can be graphed. Key concepts include the Cartesian plane, slope, distance and midpoint formulas, and relationships between lines such as parallel, perpendicular and angles between lines based on their slopes.
This document provides examples and rules for working with exponents and polynomials. It begins by showing the step-by-step working of three multiplication problems involving exponents. It then states the product rule for exponents and the rule for raising a power to another power. The document encourages positive thinking and hard work. It ends by having the reader say a phrase aloud together to reinforce a growth mindset towards math.
This document contains a math lesson on the midpoint formula. It begins with examples of using the midpoint formula to find the coordinates of the midpoint of a line segment given the coordinates of the endpoints. It then provides practice problems for students to find midpoints and missing endpoint coordinates. The document aims to teach students how to use the midpoint formula to solve geometry problems.
This learner's module discusses and help the students about the topic Systems of Linear Inequalities. It includes definition, examples, applications of Systems of Linear Inequalities.
This document contains a quiz with math problems of varying difficulty levels: easy, average, and difficult. The easy problems are worth 1 point each and cover topics like the intersection of lines, factoring, algebraic expressions, evaluating expressions, and identifying quadrants and graphs. The average problems are worth 3 points each and involve factoring quadratics, simplifying expressions, and operations with monomials. The difficult problems are worth 5 points each and require solving systems of equations, finding areas of rectangles, identifying mathematicians, and identifying perfect square trinomials. An answer key is provided with the solutions.
This document discusses analytic geometry and ellipses. It begins with objectives of defining key terms of conic sections like ellipses, and solving equations and real-world problems involving ellipses. An activity is described where students can draw an ellipse using a rubber band stretched between two pins. Examples of completed ellipses are shown along with their equations and key properties labeled like foci, vertices, axes, and eccentricity. Students are given practice problems to find properties of ellipses based on given information.
An ellipse is defined as the set of all points where the sum of the distances from two fixed points (foci) is a constant (the length of the major axis). Key properties include:
- The vertices are the endpoints of the major axis.
- The distance from the center to each focus is the eccentricity.
- The general equation of an ellipse with center at (h,k) is (x-h)^2/a^2 + (y-k)^2/b^2 = 1.
- Examples are provided to illustrate finding the equation of an ellipse given properties like the foci, vertices, or axes.
The document discusses relations and functions in mathematics. It provides an overview of key concepts to be covered, including set-builder notation, the rectangular coordinate system, and different ways to represent relations and functions using tables, mappings, graphs and rules. The objectives are for students to understand these concepts and be able to identify, illustrate, and determine different types of relations and functions.
This document discusses four methods for graphing linear equations on a coordinate plane:
1. Using any two points on the line.
2. Using the x-intercept and y-intercept.
3. Using the slope and y-intercept.
4. Using the slope and one known point.
Examples are provided to illustrate each method. Graphing linear equations is important for visualizing relationships between variables in real-life situations.
The document discusses rules for simplifying radical expressions. It states the square root and multiplication rules, which are that the square root of a squared term is the term itself, and that the square root of a product is the product of the square roots. Examples are provided to demonstrate applying these rules to simplify radical expressions by extracting square factors from the radicand. The division rule for radicals is also stated.
1. The document discusses various forms of linear equations including standard form (Ax + By = C), slope-intercept form (y = mx + b), and point-slope form (y - y1 = m(x - x1)).
2. It provides examples of transforming linear functions between these forms and discusses how to find the slope and y-intercept of a linear function from its graph.
3. The slope and direction of a linear function's graph is determined by whether its slope is positive or negative. A positive slope produces an upward rising line while a negative slope produces a downward falling line.
The document contains 10 multiple choice questions about quadratic equations. It assesses the test taker's understanding of key concepts like identifying quadratic equations, graphing quadratic functions, writing quadratic equations in standard form, and solving quadratic equations by extracting square roots. The questions range from easy to difficult levels of difficulty.
This document contains instructions and examples for working with quadratic equations. It includes directions to find the roots of quadratic equations using any method, as well as assignments involving determining values of a, b, and c in quadratic equations; finding the sum and product of roots; and analyzing whether a quadratic equation can be determined from its roots or the sum and product of roots.
This document defines slope and discusses how to calculate it using the rise over run method. Slope is the ratio of vertical change to horizontal change between two points on a line. It provides examples of finding the slopes of various lines by calculating rise over run. The document concludes by explaining that if you know the slope and one point, you can draw the line. It provides examples of drawing lines given the slope and a point.
System of Linear inequalities in two variablesAnirach Ytirahc
This document provides instructions for solving systems of linear inequalities in two variables by graphing. It defines a system of inequalities and explains that the solution is the region where the graphs of the inequalities overlap. A step-by-step process is outlined: 1) graph each inequality individually, 2) shade the appropriate half-plane, 3) the overlapping shaded regions represent the solution. An example system is graphed to demonstrate. Students will evaluate by being assigned a system to graph and answer related questions about the solution region.
This powerpoint presentation discusses or talks about the topic or lesson Direct Variations. It also discusses and explains the rules, concepts, steps and examples of Direct Variations.
This document discusses arcs and central angles in circles. It defines arcs as curved lines formed when two sides of a central angle meet at the center of a circle. There are three types of arcs: minor arcs are inside the central angle and measure less than 180 degrees; major arcs are outside the central angle; and semicircles measure 180 degrees. The measure of an arc depends on its type and the measure of the corresponding central angle. Rules are provided for calculating arc measures using central angles and properties of adjacent arcs. Examples demonstrate finding arc measures using these rules and properties of circles.
Cartesian Coordinate Plane - Mathematics 8Carlo Luna
This document explains the Cartesian coordinate plane. It describes how the plane is divided into four quadrants by the x and y axes which intersect at the origin. It provides examples of plotting points using ordered pairs with coordinates (x,y). The document also notes that Rene Descartes developed this system by combining algebra and geometry. It includes an activity for students to physically position themselves on the x and y axes to learn the coordinate system.
The document summarizes a 10 minute lesson plan for a Grade 11 mathematics class on solving quadratic inequalities. The lesson plan involves introducing quadratic inequalities, explaining the three methods for solving them, working through an example problem, having students practice solving additional problems while peer-assessing each other's work, and concluding by assessing student understanding through class work. The teacher will use a chalkboard to explain the content while students complete practice problems individually and provide peer feedback.
1. The document introduces analytic geometry and its use of Cartesian coordinate systems to determine properties of geometric figures algebraically.
2. It defines key concepts like directed lines and rectangular coordinates, and explains how to find the distance between two points and the area of polygons using their coordinates.
3. Formulas are provided to calculate distances between horizontal, vertical and slanted line segments, as well as the area of triangles and general polygons from the coordinates of their vertices. Sample problems demonstrate applying these formulas.
This document discusses parallel and perpendicular lines in the coordinate plane. It defines slope as rise over run and provides examples of calculating slope. It states that two non-vertical lines are parallel if they have the same slope. Two non-vertical lines are perpendicular if the product of their slopes is -1. Vertical and horizontal lines are always perpendicular. The objectives are to find slopes of lines and use slopes to identify and write equations for parallel and perpendicular lines.
Problem solving involving polynomial functionMartinGeraldine
The document provides examples of solving geometry problems involving finding dimensions of objects given certain constraints. The first example involves finding the size of a wood sheet needed to construct a wooden tray given its volume. The second example involves finding the lengths of the legs of a right triangle given its area and the relationship between the legs. Both examples involve setting up equations based on the given information and solving using algebraic steps to find the desired dimensions.
Analytic geometry introduced in the 1630s by Descartes and Fermat uses algebraic equations to describe geometric figures on a coordinate system. It connects algebra and geometry by plotting points using a coordinate system with real number coordinates. This allows geometric shapes to be represented by algebraic equations which can be graphed. Key concepts include the Cartesian plane, slope, distance and midpoint formulas, and relationships between lines such as parallel, perpendicular and angles between lines based on their slopes.
This document provides examples and rules for working with exponents and polynomials. It begins by showing the step-by-step working of three multiplication problems involving exponents. It then states the product rule for exponents and the rule for raising a power to another power. The document encourages positive thinking and hard work. It ends by having the reader say a phrase aloud together to reinforce a growth mindset towards math.
This document contains a math lesson on the midpoint formula. It begins with examples of using the midpoint formula to find the coordinates of the midpoint of a line segment given the coordinates of the endpoints. It then provides practice problems for students to find midpoints and missing endpoint coordinates. The document aims to teach students how to use the midpoint formula to solve geometry problems.
This learner's module discusses and help the students about the topic Systems of Linear Inequalities. It includes definition, examples, applications of Systems of Linear Inequalities.
This document contains a quiz with math problems of varying difficulty levels: easy, average, and difficult. The easy problems are worth 1 point each and cover topics like the intersection of lines, factoring, algebraic expressions, evaluating expressions, and identifying quadrants and graphs. The average problems are worth 3 points each and involve factoring quadratics, simplifying expressions, and operations with monomials. The difficult problems are worth 5 points each and require solving systems of equations, finding areas of rectangles, identifying mathematicians, and identifying perfect square trinomials. An answer key is provided with the solutions.
This document discusses analytic geometry and ellipses. It begins with objectives of defining key terms of conic sections like ellipses, and solving equations and real-world problems involving ellipses. An activity is described where students can draw an ellipse using a rubber band stretched between two pins. Examples of completed ellipses are shown along with their equations and key properties labeled like foci, vertices, axes, and eccentricity. Students are given practice problems to find properties of ellipses based on given information.
An ellipse is defined as the set of all points where the sum of the distances from two fixed points (foci) is a constant (the length of the major axis). Key properties include:
- The vertices are the endpoints of the major axis.
- The distance from the center to each focus is the eccentricity.
- The general equation of an ellipse with center at (h,k) is (x-h)^2/a^2 + (y-k)^2/b^2 = 1.
- Examples are provided to illustrate finding the equation of an ellipse given properties like the foci, vertices, or axes.
The document discusses relations and functions in mathematics. It provides an overview of key concepts to be covered, including set-builder notation, the rectangular coordinate system, and different ways to represent relations and functions using tables, mappings, graphs and rules. The objectives are for students to understand these concepts and be able to identify, illustrate, and determine different types of relations and functions.
This document discusses four methods for graphing linear equations on a coordinate plane:
1. Using any two points on the line.
2. Using the x-intercept and y-intercept.
3. Using the slope and y-intercept.
4. Using the slope and one known point.
Examples are provided to illustrate each method. Graphing linear equations is important for visualizing relationships between variables in real-life situations.
The document discusses rules for simplifying radical expressions. It states the square root and multiplication rules, which are that the square root of a squared term is the term itself, and that the square root of a product is the product of the square roots. Examples are provided to demonstrate applying these rules to simplify radical expressions by extracting square factors from the radicand. The division rule for radicals is also stated.
1. The document discusses various forms of linear equations including standard form (Ax + By = C), slope-intercept form (y = mx + b), and point-slope form (y - y1 = m(x - x1)).
2. It provides examples of transforming linear functions between these forms and discusses how to find the slope and y-intercept of a linear function from its graph.
3. The slope and direction of a linear function's graph is determined by whether its slope is positive or negative. A positive slope produces an upward rising line while a negative slope produces a downward falling line.
The document contains 10 multiple choice questions about quadratic equations. It assesses the test taker's understanding of key concepts like identifying quadratic equations, graphing quadratic functions, writing quadratic equations in standard form, and solving quadratic equations by extracting square roots. The questions range from easy to difficult levels of difficulty.
This document contains instructions and examples for working with quadratic equations. It includes directions to find the roots of quadratic equations using any method, as well as assignments involving determining values of a, b, and c in quadratic equations; finding the sum and product of roots; and analyzing whether a quadratic equation can be determined from its roots or the sum and product of roots.
This document defines slope and discusses how to calculate it using the rise over run method. Slope is the ratio of vertical change to horizontal change between two points on a line. It provides examples of finding the slopes of various lines by calculating rise over run. The document concludes by explaining that if you know the slope and one point, you can draw the line. It provides examples of drawing lines given the slope and a point.
System of Linear inequalities in two variablesAnirach Ytirahc
This document provides instructions for solving systems of linear inequalities in two variables by graphing. It defines a system of inequalities and explains that the solution is the region where the graphs of the inequalities overlap. A step-by-step process is outlined: 1) graph each inequality individually, 2) shade the appropriate half-plane, 3) the overlapping shaded regions represent the solution. An example system is graphed to demonstrate. Students will evaluate by being assigned a system to graph and answer related questions about the solution region.
This powerpoint presentation discusses or talks about the topic or lesson Direct Variations. It also discusses and explains the rules, concepts, steps and examples of Direct Variations.
This document discusses arcs and central angles in circles. It defines arcs as curved lines formed when two sides of a central angle meet at the center of a circle. There are three types of arcs: minor arcs are inside the central angle and measure less than 180 degrees; major arcs are outside the central angle; and semicircles measure 180 degrees. The measure of an arc depends on its type and the measure of the corresponding central angle. Rules are provided for calculating arc measures using central angles and properties of adjacent arcs. Examples demonstrate finding arc measures using these rules and properties of circles.
Cartesian Coordinate Plane - Mathematics 8Carlo Luna
This document explains the Cartesian coordinate plane. It describes how the plane is divided into four quadrants by the x and y axes which intersect at the origin. It provides examples of plotting points using ordered pairs with coordinates (x,y). The document also notes that Rene Descartes developed this system by combining algebra and geometry. It includes an activity for students to physically position themselves on the x and y axes to learn the coordinate system.
The document summarizes a 10 minute lesson plan for a Grade 11 mathematics class on solving quadratic inequalities. The lesson plan involves introducing quadratic inequalities, explaining the three methods for solving them, working through an example problem, having students practice solving additional problems while peer-assessing each other's work, and concluding by assessing student understanding through class work. The teacher will use a chalkboard to explain the content while students complete practice problems individually and provide peer feedback.
1. The document introduces analytic geometry and its use of Cartesian coordinate systems to determine properties of geometric figures algebraically.
2. It defines key concepts like directed lines and rectangular coordinates, and explains how to find the distance between two points and the area of polygons using their coordinates.
3. Formulas are provided to calculate distances between horizontal, vertical and slanted line segments, as well as the area of triangles and general polygons from the coordinates of their vertices. Sample problems demonstrate applying these formulas.
This document discusses parallel and perpendicular lines in the coordinate plane. It defines slope as rise over run and provides examples of calculating slope. It states that two non-vertical lines are parallel if they have the same slope. Two non-vertical lines are perpendicular if the product of their slopes is -1. Vertical and horizontal lines are always perpendicular. The objectives are to find slopes of lines and use slopes to identify and write equations for parallel and perpendicular lines.
This document provides objectives and instructions for integrating various types of functions, including:
- Rational functions using the Log Rule for Integration
- Exponential functions
- Trigonometric functions and their powers
- Functions involving inverse trigonometric functions
- Hyperbolic functions and inverse hyperbolic functions
It also gives formulas and methods for integrating specific combinations of trigonometric, exponential, and other elementary functions.
O documento discute como anfíbios são bons indicadores ambientais devido à sua sensibilidade às mudanças ambientais durante suas vidas. Apresenta estudos que mostraram o declínio de populações de anfíbios devido ao aumento dos raios UV causado pelo buraco na camada de ozônio, mas também destaca a importância de diferenciar flutuações naturais de populações de declínios antropogênicos. Relata ainda um estudo que encontrou aumento da diversidade de anuros em áreas desmatadas da Amazônia.
The document provides information about conic sections, specifically circles. It defines a circle as the set of points equidistant from a fixed point, and provides the standard equation (x - h)2 + (y - k)2 = r2, where (h, k) is the circle's center and r is the radius. Several example problems are worked through, finding the equation of circles given properties like specific points or tangency to lines. The concept of a family of circles is introduced, where the circles share a common property like center location. Radical axes are defined as the line perpendicular to the line joining two circle centers.
Here are the steps to solve this revision exercise:
1. Use the distance formula to find the lengths of each side:
a) AB = √(−1 − 2)2 + (4 − 3)2 = 3.16
b) BC = √(1 − (−1))2 + (−3 − 4)2 = 7.28
c) AC = √(1 − 2)2 + (−3 − 3)2 = 6.08
2. Use the midpoint formula to find the midpoints of each side:
a) MPAB = (0.5, 3.5)
b) MPBC = (0, 0.5)
c) MPAC = (
Here is the work to find the lateral and total surface area of the square pyramid:
The base of the pyramid is a square with sides of 6 feet. Therefore, the area of the base is:
Area of base = 6 ft × 6 ft = 36 ft2
To find the slant height, use the Pythagorean theorem:
c2 = a2 + b2
c2 = 42 + (6/2)2
c2 = 16 + 9
c2 = 25
c = 5 ft
The area of each triangular face is:
Area of one face = 1/2 × base × height
= 1/2 × 6 ft × 5 ft
=
This document provides examples and solutions for solving various types of coordinate geometry problems. It covers changing coordinate equations into standard form, writing equations given a center and radius, finding perpendicular distance between points and lines, and solving systems of linear equations graphically and algebraically. Methods demonstrated include substitution, elimination, and graphing lines to find their point of intersection.
1. The document provides 6 problems involving coordinate geometry. The problems involve finding equations of lines, points of intersection of lines, perpendicular and parallel lines, loci of points, and calculating areas of triangles. Detailed solutions and working are provided for each problem.
2. Additional problems involve finding coordinates of points based on ratios of line segments, perpendicular lines, and loci of points that satisfy given distance conditions from other points. Solutions find equations of lines and loci, and use intersections to determine coordinates.
3. The final problem calculates the area of a triangle given the coordinates of its vertices, which were previously determined based on a locus condition for one of the points.
This lesson plan aims to teach students how to use trigonometric ratios to solve problems. Students will first review trigonometric ratios, then practice solving application problems in groups. Examples include finding missing side lengths and angles of triangles. Teachers will monitor student progress and provide help as needed. To assess understanding, students will complete sample problems and discuss their work. The goal is for students to learn how to approach word problems using trigonometric concepts.
This document discusses dividing line segments into equal parts and finding points of division. It defines the median of a triangle as the line segment joining a vertex to the midpoint of the opposite side. It provides examples of finding division points, midpoints, and points that are a certain fraction of the distance between two points. Finally, it gives a triangle and asks to find the points on each median that are two-thirds of the distance from the vertex to the midpoint of the opposite side.
This document provides definitions and concepts related to analytic geometry. It discusses the Cartesian coordinate system, ordered pairs, axes, and coordinates. It defines distance formulas for horizontal, vertical, and slant line segments. Sample problems are provided to calculate distances between points and to determine geometric properties related to triangles and rectangles on a coordinate plane. The objectives are to familiarize students with the coordinate system and to determine distances, slopes, angles of inclination for lines and line segments.
This document presents information about statistics from the perspective of students in class IX E. It begins with definitions of statistics and data. Statistics is defined as the scientific study of collecting, organizing, and analyzing data to draw conclusions. There are two main types of data: qualitative and quantitative. The document then discusses key aspects of statistics including collecting and ordering data, measures of central tendency (mean, median, mode), and ways of presenting data through tables, charts and graphs. It provides examples of observing data from classes VII C and VII G on number of siblings and class rank. The summary aims to highlight the main topics and examples covered in the original lengthy document.
This document provides an overview of 10 lessons covering game mathematics topics. Lesson 6 introduces analytic geometry concepts like points, lines, planes and ellipses which are important for game development areas like rendering and collision detection. Lesson 7 covers vector mathematics including operations, applications to planes and distance calculations. Lesson 8 introduces matrices and their operations and uses for transformations. Lesson 9 covers quaternion mathematics and its application to arbitrary axis rotations. Lesson 10 applies analytic geometry concepts to areas like collision detection and reflections.
Detailed Slope Stability Analysis and Assessment of the Original Carsington E...Dr.Costas Sachpazis
A 1225 m long, 35 m high zone earth filled embankment was being constructed from 1981 to 1984 from a British Regional Water Authority to regulate flows in the River Derwent in England. The Carsington Dam was planned to be one of the largest earth filled dams in Britain. Its reservoir capacity was 35 million m3 and the watertight element was Rolled Clay Core with an upstream extension of boot shaped and shoulders of compacted mudstone with horizontal drainage layers of crushed limestone about 4 metres apart and a cut-off grout curtain (Davey and Eccles, 1983).
The downstream slope was 1:2.5 and the upstream slope 1:3. Fill placing began in May 1982 and took three summers, with winter shutdowns. In August 1983 a small berm was placed at the upstream toe to compensate for a faster rate of construction. Earth filling restarted in April 1984 and was one metre below the final crest level on 4 June 1984 when the upstream slope slipped (Skempton, 1985). Observations of pore pressure and settlement were made during construction at four sections and horizontal displacements were observed from August 1983. The Carsington Dam was almost completed on 1984.
However, at the beginning of June 1984, a 400-m length of the upstream shoulder of the embankment dam slipped some 11 m and failed. At the time of the failure, embankment construction was virtually complete with the dam approaching its maximum height of 35 m. Horizontal drainage blankets were incorporated in both the upstream and the downstream shale fill shoulders. Piezometers had been installed and pore pressures were being monitored in the foundation, in the clay core, and in the shoulder fill. The failure surface passed through the boot shaped rolled clay core and a relatively thin layer of surface clay in the foundation of the dam. Investigation of the events at Carsington has made important contributions to the fundamental understanding of the behaviour of large earthworks of this type (Vaughan et al., 1989; Dounias et al., 1996).
The objective of this research is to evaluate a detailed slope stability assessment of the Carsington Earth Embankment Dam in the UK used to retain mine tailings.
By using and applying advanced geotechnical engineering analysis tools and modelling techniques the Carsington Earth Embankment Dam, which is considered a particular geotechnical structure, is analysed.
In the current detailed slope stability analyses the total and effective stress state soil properties / parameters were used, and the most critical slip circle centre according to Fellenius - Jumikis method was initially determined. Subsequently, the Carsington Earth Embankment Dam and its foundation was analysed and examined against failure by slope instability. Considerations of loading conditions which may result to instability for all likely combinations of reservoir and tailwater levels, seepage conditions, both after and during construction were made, and hence three construction and / or loading condit
A 29-year-old single woman from Puerto Rico Caquetá, Colombia provides personal details including her telephone number, family members which includes her husband and son she lives with, occupation as a student, and favorite sport and singer. Her daily routine involves sleeping, reading, showering, eating with family, and she enjoys basketball and singer Santiago Cruz while disliking unspecified things.
- The candidate has over 6 years of experience in maintenance activities including preventative, predictive, and reactive maintenance. He has certifications in non-destructive testing techniques.
- He is seeking a challenging position that allows him to utilize his academic and practical knowledge in maintenance planning, equipment monitoring, and developing positive work environments.
- The candidate has experience supervising maintenance teams and coordinating with operations and processes to plan and execute equipment inspections and repairs.
Airwil Group commenced operations in 1998 an umbrella organisation comprising many other companies.
Over the last few years, we have built an enviable reputation based on excellence, commitment and expertise. We intend to further strengthen our brand with the highest levels of innovation, product excellence and customer trust. We have a number of distinguished property development projects to our credit in upcoming towns in India. All our projects are designed for complete peace of mind for our customers.
We give primary significance to location while making decisions related to real estate. As a strategic policy, we focus on main business districts of important cities for our commercial projects for all residential projects, we ensure the best location in terms of facilities, convenience and connectivity.
Today, malls are a glittering testimony to a shining India. In keeping with the needs of the time, we also specialize in retail space. All the malls developed by us are in the heart of retail and entertainment offering clients an incredible opportunity to have their own pride of place and reach markets of their choice.
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Today, malls are a glittering testimony to a shining India. In keeping with the needs of the time, we also specialize in retail space. All the malls developed by us are in the heart of retail and entertainment offering clients an incredible opportunity to have their own pride of place and reach markets of their choice.
We at Airwil Group feel proud of venturing into unchartered territories. We have the foresight and ability to identify cities and towns with tremendous potential and it is precisely because of this, we were amongst the first real estate companies in India to come up with projects in towns such as Ujjain, Ajmer, Bhiwani, Jabalpur, and Aligarh.
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1) The document outlines the course outcomes for Calculus I with Analytic Geometry. It discusses fundamental concepts like analytic geometry, functions, limits, continuity, derivatives, and their applications.
2) The course aims to teach students to analyze and solve problems involving lines, circles, conics, transcendental functions, derivatives, tangents, normals, maxima/minima, and related rates.
3) The assessment tasks and grading criteria are also presented, including quizzes, classwork, and a final examination. Minimum averages for satisfactory performance are provided.
This document provides an overview of key concepts in analytic geometry including:
1) Defining the inclination and slope of a single line, parallel lines, perpendicular lines, and intersecting lines.
2) Explaining that parallel lines have equal slopes and perpendicular lines have slopes that are negative reciprocals of each other.
3) Describing how to find the angle between two intersecting lines using their slopes.
This document provides an introduction to coordinate geometry and the Cartesian coordinate system. It defines key terms like coordinates, quadrants, and plotting points. The Cartesian plane is formed by the intersection of the x and y axes, with the origin at (0,0). Any point can be uniquely identified using an ordered pair (x,y) representing the distances from the x and y axes. Examples are given of plotting points and calculating distances between points on the plane using their coordinates. In summary, the document outlines the basic concepts of the Cartesian coordinate system used in coordinate geometry.
The document discusses key concepts about linear equations in two variables including:
1) It describes the Cartesian coordinate plane and how to plot points based on their x and y coordinates.
2) It explains how to find the slope, y-intercept, and x-intercept of a linear equation graphically and algebraically.
3) It provides examples of rewriting linear equations in slope-intercept form (y=mx+b) and using intercepts and slopes to graph lines on the coordinate plane.
The document discusses formulas for calculating distance, midpoints, and slopes of lines on a coordinate plane. It defines key terms like x-axis, y-axis, origin, and introduces the distance, midpoint, and slope formulas. Examples are provided to demonstrate calculating distances and slopes between points and finding midpoints, and describing lines based on whether their slopes are positive, negative, undefined, or zero.
This document provides an overview of analytic geometry concepts including:
- The Cartesian plane and using coordinates to locate points
- Equations for lines including vertical, horizontal, and oblique lines
- Geometric loci such as the midpoint and bisector
- Equations for circles, parabolas, ellipses, and hyperbolas
It includes examples of finding equations that model geometric shapes on the Cartesian plane.
History,applications,algebra and mathematical form of plane in mathematics (p...guesta62dea
The document provides information about planes and equations of planes. It defines a plane as a flat surface that extends indefinitely in width and height but has no thickness. Various plane shapes and their area formulas are described. Different forms of equations for a straight line including slope-intercept, point-slope, two-point, and standard forms are derived from the general linear equation. Two and three-dimensional Cartesian coordinate systems are also explained.
This document discusses conic sections, which are plane curves formed by the intersection of a cone with a plane. The four types of conic sections are circles, ellipses, parabolas, and hyperbolas. It provides examples of how to determine the equation of a conic section based on given properties, such as all points being equidistant from a fixed point and line. Key aspects covered include using the distance formula and completing the square to put equations in standard form.
This document provides information about coordinate grids, ordered pairs, and formulas in coordinate geometry. It defines key terms like coordinates, quadrants, and distance and section formulas. The distance formula calculates the distance between two points with coordinates (x1, y1) and (x2, y2). The section formula finds the coordinates of a point that divides a line segment between (x1, y1) and (x2, y2) in a given ratio. It also discusses finding the midpoint and calculating the area of a triangle using coordinates.
1) The document provides a refresher on analytic geometry concepts including the Cartesian plane, lines, parabolas, ellipses, and circles. It gives definitions, properties, and equations for these concepts.
2) Examples are worked through, such as finding the coordinates of points, slopes of lines, and equations of lines and circles. Practice problems and their solutions are also provided.
3) Key topics covered include the Cartesian plane, distance between points, slope and equations of lines, parallel and perpendicular lines, conic sections including parabolas, circles, and ellipses, and their defining properties and equations.
The document discusses geometric and analytical thinking. It begins by defining analytical geometry as the science that combines algebra and geometry to describe geometric figures from both algebraic and geometric viewpoints. It then discusses how analytical geometry originated with René Descartes' use of the Cartesian plane. Several geometric figures are then analyzed, including lines, circles, ellipses, and parabolas. Their key parameters and equations are defined. In particular, it provides the canonical equations for circles, ellipses, and parabolas, and discusses topics like slope and parallelism for lines.
Here are the steps to solve these problems:
1. Find the slopes of the two lines:
m1 = (8-2)/(5--2) = 6/3 = 2 (slope of r)
m2 = (7-0)/(-8--2) = 7/-6 = -1 (slope of s)
The slopes are negative reciprocals, so r ⊥ s.
2. The slopes are m1 = 2 and m2 = -1/2. Since m1 × m2 = -1, the lines are perpendicular.
3. The given line has slope 3. The perpendicular line will have slope -1/3. Plug into the point-slope form
The document discusses coordinate geometry and the Cartesian plane. It defines the key terms like the x-axis, y-axis, and origin (0,0). Any point in the plane can be located using its x and y coordinates. The gradient or slope of a line is defined as the vertical distance over the horizontal distance between two points on the line. Examples are given to demonstrate how to calculate the gradient using the gradient formula and by finding the ratio of the vertical to horizontal distances.
The coordinate plane is formed by intersecting two number lines, called the x-axis and y-axis, at their zero points. The point of intersection is called the origin. To graph an inequality in two variables, graph the boundary curve and shade the region where the inequality is true. The distance formula can be used to find the distance between two points by treating it as the hypotenuse of a right triangle formed by the differences in the x and y coordinates. The midpoint formula finds the point halfway between two points by averaging the x and y coordinates. A circle is defined as all points equidistant from a center point, where the distance from the center is the radius. The standard form of a circle equation relates the
This document provides information about coordinate geometry and various geometric concepts that can be represented using a Cartesian coordinate system. It includes:
1) An introduction to coordinate geometry and Cartesian coordinate systems.
2) Equations and properties of lines, including finding slopes, angles between lines, parallel/perpendicular lines, and intersections.
3) Equations and properties of circles, including center-radius form, diameter form, tangents, and normals.
4) Worked examples and exercises on finding equations of lines and circles given information about points, slopes, radii, etc.
Diploma-Semester-II_Advanced Mathematics_Unit-IRai University
This document provides information about coordinate geometry and various geometric concepts in a coordinate plane. It includes:
1) An introduction to coordinate geometry and the Cartesian coordinate system.
2) Definitions and methods for finding equations of lines, circles, and their relationships like parallel/perpendicular lines and tangents to a circle.
3) Worked examples and exercises for students to practice finding equations of lines and circles given information about their properties or points on them.
The document is a lesson plan on coordinate geometry for a Diploma-level course, covering topics like lines, circles, their intersections and relationships between shapes in a 2-dimensional coordinate plane.
This lecture discusses distance, midpoint, slope, lines, symmetries of graphs, equations of circles, and quadratic equations. It defines distance as the square root of the sum of the squared differences of x- and y-coordinates between two points. The midpoint formula finds the midpoint of a line segment between two points. Slope is defined as the rise over the run between two points on a line. Lines can be written in point-slope form, slope-intercept form, and intercept forms. Parallel and perpendicular lines are identified based on equal or negative reciprocal slopes. Symmetries of graphs include reflections across the x-axis, y-axis, or origin. The equation of a circle is given by (x-h)2
Distinguish equations representing the circles and the conics; use the properties of a particular geometry to sketch the graph in using the rectangular or the polar coordinate system. Furthermore, to be able to write the equation and to solve application problems involving a particular geometry.
The document provides an overview of the contents and references for the Mathematics-I course. The contents include topics like ordinary differential equations, linear differential equations, mean value theorems, functions of several variables, curvature, evolutes, envelopes, curve tracing, integration, multiple integrals, series and sequences, vector differentiation and integration. It also lists several textbooks and references for further study.
This document discusses concepts related to calculus including functions, limits, continuity, and derivatives. The objective is for students to be able to evaluate limits and determine derivatives of algebraic functions. It defines functions and function notation. It discusses limits, continuity, and the definition of the derivative. It provides examples of evaluating limits using theorems and the squeeze principle. It also defines types of discontinuities and conditions for continuity.
The document discusses the negative impacts of Spanish colonial rule in the Philippines. It summarizes that the Spanish regarded Filipinos as inferior despite introducing Christianity, enacted discriminatory laws, imposed abusive labor systems, and had a corrupt government dominated by friars. This led to Filipinos having their rights denied and being brutalized without access to fair justice. The heroic struggles of Philippines icons were needed to achieve freedom and fight for just treatment regardless of race.
Time rate problems involve quantities that change over time. Calculus is used to solve these types of problems by determining rates of change, finding maximum and minimum values, and calculating accumulated changes over time intervals. The key is setting up and solving differential equations that relate quantities and their rates of change with respect to time.
This research aimed to determine the effects of electronic cigarettes on users' health. Ten respondents, including students, vendors, salesmen and bystanders, were interviewed. The findings showed that electronic cigarettes pose health risks due to their chemical components, increasing users' risk of diseases like lung cancer and emphysema. It was concluded that electronic cigarettes cannot replace conventional cigarettes safely. The research recommended further studying electronic cigarettes' risks and potentially banning their sale if found to endanger public health.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
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How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
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Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
5. OBJECTIVE:
At the end of the lesson, the students should be able to
illustrate properly and solve application problems involving
distance formula.
6. • Analytic Geometry – is the branch of mathematics, which
deals with the properties, behaviours, and solution of points,
lines, curves, angles, surfaces and solids by means of algebraic
methods in relation to a coordinate system(Quirino and
Mijares) .
• It is a unified algebra and geometry dealing with the study of
relationships between different geometric figures and
equations by means of the geometric properties and
processes of algebra in relation to a coordinate system
( Marquez, et al).
DEFINITION:
FUNDAMENTAL CONCEPTS
7. Two Parts of Analytic Geometry
1. Plane Analytic Geometry – deals with figures on a
plane surface (two-dimensional geometry, 2D).
2. Solid Analytic Geometry – deals with solid figures
( three-dimensional geometry, 3D).
8. Directed Line – a line in which one direction is chosen as
positive and the opposite direction as negative.
Directed Line Segment – portion of a line from one point
to another.
Directed Distance – the distance from one point to
another; may be positive or negative depending upon
which direction is denoted positive.
DEFINITION:
9. RECTANGULAR COORDINATES
A pair of number (x, y) in which x is the first and y the
second number is called an ordered pair. It defines the
position of a point on a plane by defining the directed
distances of the point from a vertical line and from a
horizontal line that meet at a point called the origin, O.
The x-coordinate of a point , known also as its abscissa, is
the directed distance of the point from the vertical axis, y-
axis; while the y-coordinate, also known as the ordinate, is
its directed distance from the horizontal axis, the x-axis.
10. DISTANCE BETWEEN TWO POINTS
The horizontal distance between any two points is the
difference between the abscissa (x-coordinate) of the
point on the right minus the abscissa (x-coordinate) of
the point on the left; that is,
Horizontal Distance Between Points
Distance, d= xright − xleft
11. Vertical Distance Between Any Two Points
The vertical distance between any two points is the
difference between the ordinate (y-coordinate) of the
upper point minus the ordinate (y-coordinate) of the
lower point; that is,
Distance d = yupper − ylower
12. Distance Between Any Two Points on a Plane
The distance between any two points on a plane
is the square root of the sum of the squares of
the difference of the abscissas and of the
difference of the ordinates of the points. That is,
if
distance d = x2 − x1( )
2
+ y2 − y1( )
2
P1 x1, y1( ) and P2 x2, y2( ) are the points, then
13. SAMPLE PROBLEMS
• By addition of line segments verify whether the points A ( - 3, 0 ) ,
B(-1, -1) and C(5, -4) lie on a straight line.
• The vertices of the base of an isosceles triangle are at (1, 2) and
(4, -1). Find the ordinate of the third vertex if its abscissa is 6.
3. Find the radius of a circle with center at (4, 1), if a chord of length 4
is bisected at (7, 4).
1. Show that the points A(-2, 6), B(5, 3), C(-1, -11) and D(-8, -8) are the
vertices of a rectangle.
2. The ordinate of a point P is twice the abscissa. This point is
equidistant from (-3, 1) and (8, -2). Find the coordinates of P.
6. Find the point on the y-axis that is equidistant from (6, 1) and (-2,
-3).
15. OBJECTIVE:
At the end of the lesson, the students should be able to
illustrate properly and solve problems involving division of
line segments.
16. Let us consider a line segment bounded by the points
. This line segment can be subdivided
in some ratio and the point of division can be determined. It
is also possible to determine terminal point(s) whenever the
given line segment is extended beyond any of the given
endpoints or beyond both endpoints . If we consider the
point of division/ terminal point to be P (x, y ) and define
the ratio, r, to be
then the coordinates of point P are given by:
P1 x1, y1( ) and P2 x2, y2( )
r =
P1P
→
P1P2
→
x = x1 + r x2 − x1( )
y = y1 + r y2 − y1( )
17. If the line segment is divided into two equal parts, then the
point of division is called the midpoint. The ratio, r, is equal
to ½ and the coordinates of point P are given by:
or simply by:
x = x1 +
1
2
x2 − x1( )
y = y1 +
1
2
y2 − y1( )
x =
1
2
x1 + x2( )
y =
1
2
y1 + y2( )
18. SAMPLE PROBLEMS
•Find the midpoint of the segment joining (7, -2) and (-3, 5).
•The line segment joining (-5, -3) and (3, 4) is to be divided into five equal
parts. Find all points of division.
•The line segment from (1, 4) to (2, 1) is extended a distance equal to twice
its length. Find the terminal point.
•On the line joining (4, -5) to (-4, -2), find the point which is three-seventh the
distance from the first to the second point.
•Find the trisection points of the line joining (-6, 2) and (3, 8).
•What are the lengths of the segments into which the y-axis divided the
segment joining ( -6, -6) and (3, 6)?
•The line segment joining a vertex of a triangle and the midpoint of the
opposite side is called the median of the triangle. Given a triangle whose
vertices are A(4,-4), B(10, 4) and C(2, 6), find the point on each median that is
two-thirds of the distance from the vertex to the midpoint of the opposite
side.
20. OBJECTIVES:
At the end of the lesson, the students should be able
to use the concept of angle of inclination and slope of a line
to solve application problems.
21. INCLINATION AND SLOPE OF A LINE
The angle of inclination of the line L or simply
inclination , denoted by , is defined as the smallest
positive angle measured from the positive direction of
the x-axis to the line.
The slope of the line, denoted by m , is defined
as the tangent of the angle of inclination; that is,
And if two points are points on
the line L then the slope m can be defined as
α
m = tanα
m = tanα =
y2 − y1
x2 − x1
P1 x1, y1( ) and P2 x2, y2( )
22. PARALLEL AND PERPENDICULAR LINES
If two lines are parallel their slope are
equal. If two lines are perpendicular, the slope of
one line is the negative reciprocal of the slope of
the other line.
If m1 is the slope of L1 and m2 is the slope
of L2 then , or
Sign Conventions:
Slope is positive (+), if the line is leaning to the right.
Slope is negative (-), if the line is leaning to the left.
Slope is zero (0), if the line is horizontal.
Slope is undefined , if the line is vertical.
m1m2 = −1.
24. SAMPLE PROBLEMS
1. Find the slope, m, and the angle of inclination of the
line through the points (8, -4) and (5, 9).
2. The line segment drawn from (x, 3) to (4, 1) is
perpendicular to the segment drawn from (-5, -6) to (4,
1). Find the value of x.
3. Find y if the slope of the line segment joining (3,
-2) to (4, y) is -3.
25. ANGLE BETWEEN TWO INTERSECTING LINES
θ
α
L1
L2
ti
it
mm1
mm
tan
+
−
=θ
Where: mi = slope of the initial side
mt = slope of the terminal side
The angle between two intersecting lines is the positive angle
measured from one line (L1) to the other ( L2).
0
180:note =∠+∠ αθ
26. Sample Problems
1.Find the angle from the line through the points (-1, 6)
and (5, -2) to the line through (4, -4) and (1, 7).
2.The angle from the line through (x, -1) and (-3, -5) to
the line through (2, -5) and (4, 1) is 450
. Find x
3.Two lines passing through (2, 3) make an angle of 450
with one another. If the slope of one of the lines is 2,
find the slope of the other.
27. AREA OF A POLYGON BY COORDINATES
Consider the triangle whose vertices are P1(x1, y1), P2(x2, y2)
and P3(x3, y3) as shown below. The area of the triangle can
be determined on the basis of the coordinates of its
vertices.
o
y
x
( )111 y,xP
( )222 y,xP
( )333 y,xP
28. Label the vertices counterclockwise and evaluate the area
of the triangle by:
1yx
1yx
1yx
2
1
A
33
22
11
=
The area is a directed area. Obtaining a negative value
will simply mean that the vertices were not named
counterclockwise. In general, the area of an n-sided
polygon can be determined by the formula :
1n54321
1n54321
yy..yyyyy
xx..xxxxx
2
1
A =
29. SAMPLE PROBLEMS
1. Find the area of the triangle whose vertices are (-
6, -4), (-1, 3) and (5, -3).
2.Find the area of a polygon whose vertices are (6,
-3), (3, 4), (-6, -2), (0, 5) and (-8, 1).
31. OBJECTIVE:
At the end of the lesson, the students should be able to
determine the equation of a locus defining line, circle and
conics and other geometries defined by the given condition.
32. EQUATION OF A LOCUS
An equation involving the variables x and y is usually
satisfied by an infinite number of pairs of values of x and y,
and each pair of values corresponds to a point. These points
follow a pattern according to the given equation and form a
geometric figure called the locus of the equation.
Since an equation of a curve is a relationship satisfied
by the x and y coordinates of each point on the curve (but
by no other point), we need merely to consider an arbitrary
point (x,y) on the curve and give a description of the curve
in terms of x and y satisfying a given condition.
33. Sample Problems
Find an equation for the set of all points (x, y) satisfying
the given conditions.
1. It is equidistant from (5, 8) and (-2, 4).
2.The sum of its distances from (0, 4) and (0, -4) is 10.
3.It is equidistant from (-2, 4) and the y-axis
35. OBJECTIVE:
At the end of the lesson, the students is
should be able to write the equation of a line in the
general form or in any of the standard forms; as
well as, illustrate properly and solve application
problems concerning the normal form of the line.
36. STRAIGHT LINE
A straight line is the locus of a point that
moves in a plane in a constant slope.
Equation of Vertical/ Horizontal Line
If a straight line is parallel to the y-axis
( vertical line ), its equation is x = k, where k is
the directed distance of the line from the y-axis.
Similarly, if a line is parallel to the x-axis
( horizontal line ), its equation is y = k, where k is
the directed distance of the line from the x-axis.
37. General Equation of a Line
A line which is neither vertical nor horizontal
is defined by the general linear equation
Ax + By + C=0,
where A and B are nonzeroes.
The line has y-intercept of and slope of .−
C
B
−
A
B
38. DIFFERENT STANDARD FORMS OF THE
EQUATION OF A STRAIGHT LINE
A. POINT-SLOPE FORM:
If the line passes through the points ( x , y) and (x1, y1), then the
slope of the line is .
Rewriting the equation we have
which is the standard equation of the point-slope form.
1
1
xx
yy
m
−
−
=
( )11 xxmyy −=−
39. B. TWO-POINT FORM:
If a line passes through the points (x1, y1) and (x2, y2), then the
slope of the line is .
Substituting it in the point-slope formula will result to
the standard equation of the two-point form.
12
12
xx
yy
m
−
−
=
( )1
12
12
1 xx
xx
yy
yy −
−
−
=−
40. C. SLOPE-INTERCEPT FORM:
Consider a line containing the point P( x, y) and not parallel to
either of the coordinate axes. Let the slope of the line be m
and the y-intercept ( the intersection point with the y-axis) at
point (0, b), then the slope of the line is .
Rewriting the equation, we obtain
the standard equation of the slope-intercept form.
0x
by
m
−
−
=
bmxy +=
41. D. INTERCEPT FORM:
Let the intercepts of a line be the points (a, 0), the x-
intercept, and (0, b), the y-intercept. Then the slope of
the line is defined by .
Using the Point-slope form, the equation is written as
or simply as
the standard equation of the Intercept Form.
a
b
m −=
( )0x
a
b
by −−=−
1
b
y
a
x
=+
42. E. NORMAL FORM:
Suppose a line L, whose equation is to be found, has its
distance from the origin to be equal to p. Let the angle of
inclination of p be .
Since p is perpendicular to L, then the slope of p is equal to the
negative reciprocal of the slope of L,
Substituting in the slope-intercept form y = mx + b , we obtain
or
the normal form of the straight line
θ
θ
θ
θ
θ sin
cos
mor,cot
tan
1
m −=−=−=
θθ
θ
sin
p
x
sin
cos
y +−=
py sincosx =+ θθ
43. Reduction of the General Form to the Normal Form
The slope of the line Ax+By+C=0 is . The slope of p which is
perpendicular to the line is therefore ; thus, .
From Trigonometry, we obtain the values
and .
If we divide the general equation of the straight line by
, we have
or
This form is comparable to the normal form .
Note: The radical takes on the sign of B.
B
A
−
A
B
A
B
tan =θ
22
BA
B
sin
+±
=θ
22
BA
A
cos
+±
=θ
22
BA +± 0
BA
C
y
BA
B
x
BA
A
222222
=
+±
+
+±
+
+±
BA
C
y
BA
B
x
BA
A
222222
+±
−
=
+±
+
+±
py sincosx =+ θθ
44. PARALLEL AND PERPENDICULAR LINES
Given a line L whose equation is Ax + By + C = 0.
The line Ax + By + K = 0 , for any constant K not
equal to C, is parallel to L;
and the Bx – Ay + K = 0 is perpendicular to L.
45. DIRECTED DISTANCE FROM A LINE TO A POINT
The directed distance of the point P(x1, y1) from the
line Ax + By + C = 0 is ,
where the radical takes on the sign of B.
22
11
BA
CByAx
d
+±
++
=
47. Sample Problems
1. Determine the equation of the line passing through (2, -3) and
parallel to the line through (4,1) and (-2,2).
2. Find the equation of the line passing through point (-2,3) and
perpendicular to the line 2x – 3y + 6 = 0
3. Find the equation of the line, which is the perpendicular bisector
of the segment connecting points (-1,-2) and (7,4).
4. Find the equation of the line whose slope is 4 and passing through
the point of intersection of lines x + 6y – 4 = 0 and 3x – 4y + 2 = 0.
5. The points A(0, 0), B(6, 0) and C(4, 4) are vertices of triangles. Find:
a. the equations of the medians and their intersection point
48. 6. Find the distance from the line 5x = 2y + 6 to the point (3, -5).
7. Find the equation of the bisector of the acute angles and also the
bisector of the obtuse angles formed by the lines x + 2y – 3 = 0 and 2x + y –
4 = 0.
8. Determine the distance between the lines: 2x + 5y -10 =0 ; 4x + 10y + 25
= 0.
9. Write the equation of the line a) parallel to b) perpendicular to 4x + 3y
-10 = 0 and is 3 units from the point ( 2, -1).
49. CLASSWORK 1
1. The abscissa and ordinate of a point units from (3, 3) are
numerically equal but of opposite signs. Find the point.
2.Given two points A(8, 6) and B(–7, 9), determine a third point P(x,
y) such that the slopes of AP and BP are ½ and –2/3 respectively.
3.A line through (–6,–7) and (x, 7) is perpendicular to a line through
(1,–4) and (–5, 2). Find x.
4.A line passes through (6,–4) and makes an angle of 1350
with the
x-axis. Find the equation of the line.
5.The angle from the line through (–1, y) and (4,–7) to the line
through (4, 2) and (–1,–9) is 1350
. Find y.
6.Find the equation of the bisector of the obtuse angle between
the lines x + 2y – 3 = 0 and 2x + y – 4 = 0.
52
50. REFERENCES
Analytic Geometry, 6th
Edition, by Douglas F. Riddle
Analytic Geometry, 7th
Edition, by Gordon Fuller/Dalton Tarwater
Analytic Geometry, by Quirino and Mijares
Fundamentals of Analytic Geometry by Marquez, et al.
Algebra and Trigonometry, 7th
ed by Aufmann, et al.