Geometry Formulas and Concepts for Distance, Midpoint, and Circles
Explore the distance and midpoint formulas in geometry to calculate distances and midpoints between points, as well as understand the concepts of circles, including definitions and equations for circles. Enhance your understanding of geometric shapes and measurements.
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Chapter 10, Section 1 Distance and Midpoint Formulas; Circles Page 758
The Distance Formula ( ) ( ) x1,y1 x2,y2 The distance, d, between the points and in the rectangular coordinate system is: ( ) ( ) 2+ y2-y1 2 d= x2-x1
Example Find the distance between (-1, -3) and (2, 3). Express the answer in simplified radical form. Solution: With the formula, identify the four values needed. = (-1, 3) and = (2, 3) Place into the formula and solve ( ) ( ) 2+ y2-y1 2 d= x2-x1 ( ) ( ) x1,y1 x2,y2
continue ( ) ( ) 2+ y2-y1 2 d= x2-x1 Formula: ( ) ( ) ) x2,y2 x1,y1 Values: = (-1, -3) and = (2, 3) So d = 2- -1 ( ) ( 2+ 3- -3 ( ) ( ) 2 Simplify: d = 32+62 3 5 d = d = units 45
Your turn Find the distance between the points (2, 3) and (14, 8).
The Midpoint Formula ( ) ( ) x1,y1 x1,y2 Consider the line segment whose endpoints are and The coordinates of the segment s midpoint are x1+x2 2 ,y1+y2 2
Example Find the midpoint of a line segment with endpoints ( 1, - 6) and ( - 8, - 4) x1+x2 ,y1+y2 2 Solution: with the formula, find the indicated values. 2 ( x2,y2 ) ( ) x1,y1 1+ -? 8 ( 2 ) ( ) = (1, - 6) and = (- 8 - 4) Place the values into the formula and simplify: So the midpoint is 2,-5 ,-? 6+ -? 4 2 -7
Your turn Find the coordinates of the midpoint of a line segment that has endpoints at (- 4, -7) and (-1, -3)
Circles Definition: Set of all points in a plane that are equidistant from a fixed point, called the center. The fixed distance from the circle s center to any point on the circle is called the radius. diameter radius center
Vocabulary Center - point Radius from center to circle Diameter - endpoints on the circle passing through the center. - length is 2 times the radius. Standard Form of the Equation of a circle: Center at (h, k), radius, r ( ) 2+ y-k ( ) 2=r2 x-h General Form of the Equation of a Circle x2+y2+Dx+Ey+F=0
Example 1 Write the standard form of the equation of a circle that has a center at (-2, 3) with a radius of 5. ( ) 2+ y-k ( ) 2=r2 x-h Solution: Use the formula: where (h, k) = (-2, 3) r = 5 Substitute: x+2 ( ) 2+ y-3 ( ) 2=25
Example 2 Find the center and radius of the circle that has the equation: ( ) 2+ y+4 ( ) 2=36 x-3 Solution: Relate the values with the formula: h = 3, k = - 4, r = 6 So the center is at (3, - 4) and the radius is 6 ( ) 2+ y-k ( ) 2=r2 x-h
Example 3 x2+y2+4x-6y-23=0 Given the equation of the circle: Rewrite in standard form. Solution: Complete the square.
Problem Find the standard form of the equation of a circle that the endpoints of the diameter of a circle are (- 4, 1) and (2, 5).
Summary Find the distance between two points. ( ) ( ) 2+ y2-y1 2 d= x2-x1 Find the midpoint of a line segment given the endpoints. x1+x2 2 2 ,y1+y2 Given the center and radius of a circle, find the equation. ( ) 2+ y-k ( ) 2=r2 x-h
