Circle Equations Apr 6
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This document defines key terms and concepts related to circles and their equations. It discusses that a circle consists of all points equidistant from a fixed center point, and that the radius is the distance from the center to any point on the circle. It provides the standard equation for a circle and an alternate equation that can be converted to standard form by completing the square.
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The document discusses equations of circles and how to find the equation given the center and radius or vice versa. It provides examples of finding the equation of circles with given centers and radii, as well as finding the center and radius from a given equation. It also shows how to find the equation of a circle if the center and a point on the circle are given.
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3) More generally, for a circle with center (a,b), the equation is (x-a)^2 + (y-b)^2 = r^2, where (a,b) are the coordinates of the center and r is the radius.
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This document discusses graphing circles and converting between graphing form and standard form equations of circles. It provides examples of writing the equation of a circle given its center and radius in graphing form. It also shows how to find the center, radius, and equation of a circle given the standard form equation. The key points are:
- In graphing form, the center is denoted as (h, k) and the equation is (x - h)2 + (y - k)2 = r2
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Here are some things you did well and could improve on:
WWW:
- You explained the key concepts around writing the equation of a circle clearly and concisely. Breaking it down step-by-step makes it easy to understand.
- Providing examples with worked solutions is very helpful for reinforcement. The visual diagrams additionally aid comprehension.
- Giving practice problems for students to try on their own, along with answers, allows for application of the material.
EBI:
- Some of the text could be formatted for easier reading (e.g. consistent formatting of equations).
- Adding brief summaries or recaps after sections of explanation may aid retention.
- Providing guidance on common errors
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The document discusses topics in coordinate geometry including slopes, length of line segments, midpoints, and proofs. It provides formulas and examples for calculating slopes, lengths of lines, and midpoints. It also discusses using an analytical approach to prove geometric theorems through using coordinates, formulas, and algebraic manipulations rather than relying solely on diagrams.
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Coordinate geometry represents points on a number line with real numbers. The number line places positive numbers to the right of zero and negative numbers to the left.
The Cartesian coordinate system uniquely determines points in two or three dimensional space using perpendicular x and y axes intersecting at the origin. René Descartes developed this system, linking algebra and geometry by describing curves and lines with equations.
The coordinate plane has perpendicular x and y axes. Points are located using their x and y coordinates, which represent horizontal and vertical distances from the origin. The plane is divided into four quadrants numbered counter-clockwise from the top right. Ordered pairs notation (x,y) specifies a point's location by listing the x coordinate first.
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This document provides information on various concepts in elementary and additional mathematics including:
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- 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.
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The document defines key concepts related to coordinate grids and ordered pairs:
- A coordinate grid uses horizontal and vertical intersecting lines to locate points via distances from the lines.
- The x-axis is horizontal and the y-axis is vertical.
- Ordered pairs use the format (x,y) to give the location of a point by listing the distance from each axis.
- Points can be plotted on a four-quadrant grid by first finding the x value and then the y value, with negative numbers indicating left/below the origin.
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The document defines key terms and concepts related to circles and their equations. It explains that a circle consists of points equidistant from a fixed center point, and defines the radius as the distance from the center to any point on the circle. It provides the standard equation for a circle with center at the origin, and notes that the standard form includes variables h and k to indicate the center coordinates and r for the radius. It also describes a second form for the circle equation that can be converted to standard form by completing the square.
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This document discusses the standard and general forms of equations for circles. It provides examples of writing the standard and general form equations given the center and radius of various circles. Key points covered include:
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This document provides information about circles and their relationships to other geometric concepts:
1. It defines key concepts such as the distance formula, points on a circle, the equation of a circle with the center at the origin or other point (x,y), and the general equation of a circle.
2. It discusses the intersection of circles and lines, including finding the points of intersection using substitution and solving quadratics. The discriminant is used to determine the number of intersection points.
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The document provides a history and overview of circles. It discusses:
1) Euclid defined circles in 300 BC in his work "The Elements", establishing circles as fundamental objects in geometry. He defined a point and line and established early theorems about circles.
2) Key parts of a circle include the radius, diameter, circumference, chord, secant, tangent, arc, sector, and segment. The circumference is related to the diameter by pi.
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The document discusses equations of circles and how to find the equation given the center and radius or vice versa. It provides examples of finding the equation of circles with given centers and radii, as well as finding the center and radius from a given equation. It also shows how to find the equation of a circle if the center and a point on the circle are given.
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The document discusses similar figures and scale drawings. Similar figures have the same shape but not necessarily the same size, and have equal corresponding angles and proportional corresponding sides. Scale drawings are enlarged or reduced drawings that are similar to an actual object, with the scale ratio representing the ratio between a distance on the drawing and the actual distance. Scale drawings and similar figures can be used to indirectly measure quantities through setting up proportional relationships between known and unknown values.
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1) The document discusses the equation of a circle and how to derive it using the Pythagorean theorem and distance formula.
2) For a circle with center (0,0), the equation is x^2 + y^2 = r^2, where r is the radius.
3) More generally, for a circle with center (a,b), the equation is (x-a)^2 + (y-b)^2 = r^2, where (a,b) are the coordinates of the center and r is the radius.
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This document discusses graphing circles and converting between graphing form and standard form equations of circles. It provides examples of writing the equation of a circle given its center and radius in graphing form. It also shows how to find the center, radius, and equation of a circle given the standard form equation. The key points are:
- In graphing form, the center is denoted as (h, k) and the equation is (x - h)2 + (y - k)2 = r2
- To convert to standard form, complete the square and factor the left side, moving any constant to the right side.
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The document summarizes a school project on graphically representing different areas of their school. It includes graphs of the school boundary, playing field, basketball court, and other areas with their measurements. It then has a pie chart showing the distribution of the school area among different sections. The students enjoyed measuring, creating graphs and the project helped increase their knowledge.
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Here are some things you did well and could improve on:
WWW:
- You explained the key concepts around writing the equation of a circle clearly and concisely. Breaking it down step-by-step makes it easy to understand.
- Providing examples with worked solutions is very helpful for reinforcement. The visual diagrams additionally aid comprehension.
- Giving practice problems for students to try on their own, along with answers, allows for application of the material.
EBI:
- Some of the text could be formatted for easier reading (e.g. consistent formatting of equations).
- Adding brief summaries or recaps after sections of explanation may aid retention.
- Providing guidance on common errors
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The document discusses topics in coordinate geometry including slopes, length of line segments, midpoints, and proofs. It provides formulas and examples for calculating slopes, lengths of lines, and midpoints. It also discusses using an analytical approach to prove geometric theorems through using coordinates, formulas, and algebraic manipulations rather than relying solely on diagrams.
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Coordinate geometry represents points on a number line with real numbers. The number line places positive numbers to the right of zero and negative numbers to the left.
The Cartesian coordinate system uniquely determines points in two or three dimensional space using perpendicular x and y axes intersecting at the origin. René Descartes developed this system, linking algebra and geometry by describing curves and lines with equations.
The coordinate plane has perpendicular x and y axes. Points are located using their x and y coordinates, which represent horizontal and vertical distances from the origin. The plane is divided into four quadrants numbered counter-clockwise from the top right. Ordered pairs notation (x,y) specifies a point's location by listing the x coordinate first.
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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.
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The document defines key concepts related to coordinate grids and ordered pairs:
- A coordinate grid uses horizontal and vertical intersecting lines to locate points via distances from the lines.
- The x-axis is horizontal and the y-axis is vertical.
- Ordered pairs use the format (x,y) to give the location of a point by listing the distance from each axis.
- Points can be plotted on a four-quadrant grid by first finding the x value and then the y value, with negative numbers indicating left/below the origin.
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The document defines key terms and concepts related to circles and their equations. It explains that a circle consists of points equidistant from a fixed center point, and defines the radius as the distance from the center to any point on the circle. It provides the standard equation for a circle with center at the origin, and notes that the standard form includes variables h and k to indicate the center coordinates and r for the radius. It also describes a second form for the circle equation that can be converted to standard form by completing the square.
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The document discusses the standard form of a circle equation and how to find the center and radius from the equation. It provides examples of writing the equation of a circle given the center and/or radius or a point on the circle. The standard form is (x - h)2 + (y - k)2 = r2, where (h, k) are the coordinates of the center and r is the radius. If the equation is not in standard form, it may need to be completed into this form by completing the square.
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The document discusses the standard form of a circle equation and how to find the center and radius from the equation. It provides examples of writing the equation of a circle given the center and/or radius or a point on the circle. The standard form is (x - h)2 + (y - k)2 = r2, where (h, k) are the coordinates of the center and r is the radius. If the equation is not in standard form, it may need to be completed into a perfect square to extract the center and radius.
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The document discusses the standard form of the equation of a circle and how to find the center and radius from the equation. It provides examples of writing the equation given the center and radius, as well as finding the center and radius from an equation. Key aspects covered include:
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The document discusses the standard form of the equation of a circle and how to find the center and radius from the equation. It provides examples of writing the equation given the center and radius, as well as finding the center and radius from an equation. Key aspects covered include:
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Doug's Truck project
The document summarizes work done over two summers to restore a truck cab. It describes picking up the cab in Detroit to replace the original, taking out the old cab, sandblasting and priming the cab and frame, doing bodywork like installing a new floor, and finally lifting the restored cab onto the frame.
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Dave needs a new computer for university costing $3,294 before taxes. Financing is available at 18.99% compounded monthly over 48, 36, or 24 months. The monthly payments and total paid are calculated for each term. Daniel can also lease the same computer for $106.92/month over 48 months, $132.32/month over 36 months, or $182.88/month over 24 months. The document then provides information about leasing, including what costs make up monthly lease payments and details that would be in a lease agreement.
Lesson 2 Apr 6 2010
The document discusses various interest rates and compounding periods for investments, savings accounts, and credit cards, and uses calculations to determine future values, effective interest rates, and the best investment options based on interest rates and compounding frequencies. Formulas like the Rule of 72 are presented for estimating doubling times given interest rates.
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The monthly payments and total amount paid will increase as the loan term decreases from 48 months to 36 months to 24 months due to the interest being applied over a shorter period of time.
Lesson 1 Apr 5 2010
This document discusses personal finance concepts like the time value of money and compound interest. It provides the basic formulas for calculating future value (FV), present value (PV), interest rate (I%), number of periods (N), principal (P), payments (PMT), periodic interest rate (r), number of compounding periods per year (n), and time (t). The document works through examples of using these formulas to calculate things like how much money you will have after investing a principal amount over a period of time at a given interest rate.
Pretest
The document contains 6 math problems: 1) Find the value of k that gives equal roots of the quadratic equation f(x)=x^2 + 4x + k. 2) Determine the nature of the roots of the quadratic equation 13x^2 - 15x = 4. 3) Solve the equation x - 3 = 2 for x. 4) Find the quadratic equation with integer coefficients given the roots 3 + 9i and 10. 5) Solve the equation x + 8 = 10x - 81 for x. 6) Solve the equation x/3 = 6 - 2x for x.
Lesson 03 Appsof Integrals Mar23
This document discusses using the shell method to calculate volumes of solids generated by revolving regions between functions around axes. It provides examples of revolving the function f(x)=x^2 around the x-axis and y-axis, and revolving the region between f(x)=0.5x^2-2x+4 and g(x)=4+4x-x^2 around both the x-axis and y-axis. Instructions are given to use the shell method to find each volume.
Lesson 08 Mar 23
Absolute value refers to the distance from zero on the number line. There are two values, -2 and 2, that have an absolute value of 2. Absolute value can never be negative because distance cannot be negative. The document provides examples of using absolute value to solve equations.
Lesson 07 Mar 20
This document discusses solving rational equations by finding the least common denominator, combining like terms, and then solving the resulting equation for the variable. It contains an exercise with questions 1, 6, and 7 about solving rational equations.
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This document provides instructions and problems for solving vector problems by drawing scale diagrams, adding vectors using the triangle method, and calculating distances and directions from starting points. Specifically, it asks the reader to: 1) draw scale diagrams of vectors for a person walking 13 blocks E15°S and a boat headed 300° at 45 km/h; 2) add the vectors using the triangle method; and 3) solve problems involving distances and directions for a man walking in different directions and a jogger moving north and east over time.
Lesson 06 Mar 19
An equation containing a radical is called a radical equation. This document refers to exercises 18 questions 1 through 5 and also questions 8 and 9 which involve solving or working with radical equations. The goal is to extract the key essential information about radical equations from the given document in 3 sentences or less.
Vectors 02 Mar 19
The document provides information about vector addition and trigonometric equations. It discusses drawing scale diagrams to represent vectors and their directions and magnitudes. Methods for adding vectors are described, including the triangle method used when vectors are tip-to-tail and the parallelogram method used when vectors are tail-to-tail. An example of each method is worked out to find the resultant vector.
Lesson 02 Appsof Integrals Mar19
The document contains instructions to find the volumes of solids generated by revolving regions bounded by graphs about axes. It gives the volume as 183.981 when revolving the region between the graphs y = 2x + 4 and y = ex about the x-axis. It also gives the volumes as 7/15 when revolving the region between y = x^2 + 1 and y = x + 1 about the x-axis and 4/5 when revolving the same region about the line y = -1.
Vectors 02 Mar 19
The document provides information about vector addition and trigonometric equations. It discusses drawing scale diagrams to represent vectors and their directions and magnitudes. Specific examples are given of adding vectors using the triangle method when vectors are tip-to-tail and the parallelogram method when they are tail-to-tail. Measurements from the scale diagrams along with a protractor can be used to find the resultant vector.
Vectors 01 Mar 18
The document discusses vectors and provides examples of identifying quantities as scalar or vector. It also discusses four notations for writing vectors using arrows, bearings, angle-direction-direction, and angle-direction of direction. Examples are given to demonstrate each notation. The document also discusses stating the direction of vectors in five ways and using diagrams of parallelograms to name vectors that are equal, opposite, collinear, or parallel but not equal to other vectors in the diagram.
Lesson 05 Mar 18
The document describes a problem where a rectangular piece of cardboard has a length longer than its width. Square pieces are cut from the corners and the sides are folded up to form a box with a volume of 450 cm3. The original length of the cardboard was 16 cm and the width was 11 cm.
Lesson 04 Mar 17
This document discusses methods for finding the roots of quadratic equations. It introduces the discriminant formula to determine the type of roots, and explains how to use the quadratic formula to find the exact values of the roots. It also shows how to write a quadratic equation given the sum and product of its roots, or given two integer roots.
Lesson 01 Appsof Integrals Mar17
The document discusses calculating the volume of solids of revolution using integrals. It provides the formula for finding the volume of a solid rotated about the x-axis between x=a and x=b using a cross-sectional area function A(x). It then works through an example of finding the volume of a right circular cone of height 4 and base radius 1, and confirms the result matches the standard volume formula for a cone.
Lesson 03 mar 16
This document provides instructions to solve equations by graphing them. The reader is directed to graph two or more equations simultaneously and find the point(s) of intersection, which will represent the solution(s) to the system of equations. The document references exercise problems 15 questions 1 through 3, which likely involve graphing and solving systems of equations.
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Circle Equations Apr 6
- 1. Circles
- 2. Circle: the set of points ( x , y ) that are equidistant from a fixed point called the centre of the circle. Radius: the distance between the centre of the circle and any point ( x , y ) on the circle
- 7. Equation?