Analytic geom and distance formula
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Analytic geometry is the merging of algebra and geometry into a unified body of knowledge that occurred in the early 1600s through the work of René Descartes and Pierre de Fermat. It defines the Cartesian plane with axes and quadrants used to plot points and determine distances between points through calculation of line segments. Examples are provided to illustrate determining if points are collinear or what type of triangle is formed by three points based on side lengths.
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CLASS X MATHS
This document provides an overview of coordinate geometry. It begins by defining the Cartesian coordinate system, which uses an ordered pair of numbers to describe the position of points on a plane. It then discusses the four quadrants of the coordinate plane and explains how to find the coordinates of a point. Other topics covered include the distance formula, midpoint formula, section formula, and formulas for finding the area of triangles and determining collinearity of three points using coordinate geometry. Examples are provided to illustrate each concept. The document concludes by suggesting using coordinate geometry to mark landmarks on a city map.
AS LEVEL CIRCULAR MEASURE GEOMETRY EXPLAINED
The document defines a radian as the angle subtended by an arc of a circle equal in length to the radius. It states that π radians equals 180 degrees, so 1 radian equals about 57.29 degrees. Formulas are provided for finding the circumference of an arc and area of a sector using radians. Examples are given to practice converting between radians and degrees and using the formulas. Special angle values are listed in a table. Past exam questions are worked through, applying concepts of radians, sectors, arcs, and right triangles in semi-circles.
Pmo20072008questions&solutions(6)
The document contains information about the Tenth Philippine Mathematical Olympiad held in 2007-2008, including its aims, selection process, and organization. It provides the problems and answers from the area and national stages of the competition. The area stage contained 23 problems, while the national stage consisted of oral rounds with 15-second, 30-second, and 60-second problems testing various mathematical concepts and skills.
Plano numerico.
The document discusses various topics in geometry including:
- The Cartesian plane and locating points with coordinates like A=(6,4).
- Formulas for distance between points and finding midpoint coordinates.
- Equations for circles, parabolas, ellipses, and hyperbolas.
- Worked examples of graphing these shapes and finding their properties.
Archimedes
1. The document discusses how Archimedes may have calculated the area under a parabola and the centroid of a parabolic region without calculus, using only geometry. It estimates that Archimedes' results came close to matching what is obtained through calculus.
2. It suggests experiments Archimedes could have performed to estimate the centroid of a half-parabola by constructing cardboard models and balancing them. The results of these experiments closely matched the calculus solution.
3. It examines how Archimedes may have estimated the arc length of a parabola and spiral using measurement tools available to him like dividers, and how his work laid the foundations for later mathematicians like Pappus
Co-ordinate Geometry.pptx
Introduction to Co-ordinate Geometry
Mapping the plane
Distance between two points
Distance formula
Properties of distance
Midpoint of a line segment
Midpoint formula
Coordinate geometry 9 grade
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.
Ijetr011921
ER Publication,
IJETR, IJMCTR,
Journals,
International Journals,
High Impact Journals,
Monthly Journal,
Good quality Journals,
Research,
Research Papers,
Research Article,
Free Journals, Open access Journals,
erpublication.org,
Engineering Journal,
Science Journals,
1. ejercicios
This document contains a series of exercises related to vectors in a plane. It begins with exercises involving vector operations like finding scalar multiples that satisfy equations and vector addition and subtraction. Later questions involve vector properties such as parallelism of vectors, orthogonality, vector lengths, and linear combinations of vectors. Geometric representations of vectors are also explored through problems finding points and line segments. The document aims to reinforce concepts of vector algebra and geometry through multiple practice problems.
Trigratios
1. A famous quotation of Pythagoras is given in the boxes, with some boxes illustrating properties of right triangles and others forming the quotation.
2. Crossing out the box illustrating a property of a right triangle reveals the quotation "Above all things, reverence yourself."
3. The document then reviews key concepts about right triangles including their parts, similarity, and trigonometric ratios defined using the sine, cosine, and tangent functions.
Class 10 Cbse Maths Sample Paper Term 1 Model 3
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Core sub math_att_4pythagoreantheorem
The document provides information about the Pythagorean theorem:
1) It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
2) It gives examples of right triangles that satisfy the theorem, such as ones with sides of 3, 4, 5 or 5, 12, 13.
3) It includes an animated proof of the theorem showing how the area of the square on the hypotenuse equals the combined areas of the squares on the other two sides.
Coordinate geometry
This document provides information on various concepts in elementary and additional mathematics including:
- The distance, midpoint, and gradient formulas for lines
- Equations of lines
- Parallel and perpendicular lines
- Intersecting lines and finding intersection points
- Perpendicular bisectors
- Finding the area of polygons
It includes examples of applying these concepts to solve problems involving lines, midpoints, gradients, intersections, perpendiculars, and calculating areas.
Coordinategeometry1 1
The document discusses coordinate geometry and defines key terms like axes of reference, coordinates of a point, distance formula, section formula, and area of a triangle. It provides examples of using these concepts and formulas to solve problems like finding the coordinates of a point dividing a line segment in a given ratio, or the area of a triangle given the coordinates of its vertices.
Coordinategeometry1 1
The document discusses key concepts of coordinate geometry including: Rene Descartes introduced coordinate geometry by representing points on a plane using two numbers called coordinates; the plane is divided into four parts by perpendicular x and y axes intersecting at the origin; the coordinates of a point P are written as an ordered pair (x,y) giving the distance from the origin along the x and y axes; and the distance formula can be used to find the distance between two points on the plane.
TRIANGLE CONGRUENCE, SSS, ASA, SAS and AAS
This document discusses congruent triangles and their properties. It begins by introducing triangles and their historical uses. It then defines different types of triangles based on their sides and angles. The document explains the triangle congruence postulates of SSS, SAS, ASA, and uses examples to prove triangles congruent. It also discusses using congruent triangles to solve real-world problems involving structures. The document warns that there is no SSA or AAA postulate for triangle congruence. It concludes by discussing properties of isosceles and equilateral triangles.
Analisis unidad 3
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.
Measuring Segments and Coordinate Plane
This slideshow was used to introduce application of Segment Addition Postulate along with Coordinate Plane in Geometry. There is a review of several concepts at the end of the two lessons.
Math14 lesson 1
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.
18Ellipses-x.pptx
The document discusses conic sections and ellipses. Conic sections are graphs of quadratic equations of the form Ax2 + By2 + Cx + Dy = E, where A and B are not both 0. Their graphs include circles, ellipses, parabolas and hyperbolas. Ellipses are defined as the set of all points where the sum of the distances to two fixed foci is a constant. Ellipses have a center, two axes called the semi-major and semi-minor axes, and radii along the x and y axes called the x-radius and y-radius. The standard form of an ellipse equation is presented.
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Analytic geom and distance formula
- 1. Prepared by: Mr. Raymond B. Canlapan
- 2. FLIES
- 5. Prepared by: Mr. Raymond B. Canlapan
- 8. REVIEW: PLOT (3, -4)
- 9. Prepared by: Mr. Raymond B. Canlapan
- 10. What is the distance between the two points?
- 11. Remember:
- 17. Prepared by: Mr. Raymond B. Canlapan