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LESSON 10-4 Equations of Circles Created by Lisa Palen and Kristina Green Henrico High School Part I Equations of Circles Recall: Definitions • Circle: The set of all points that are the same distance from the center • Radius: a segment whose endpoints are the center and a point on the circle • Radius: the LENGTH of a radius Center Radi us Equation of a Circle x y r 2 2 2 x h y k 2 Center (0, 0) Radius = r 2 r 2 Center (h, k) Radius = r Finding the Center and the Radius when given the equation 1. x y 25 2 2 Center (0, 0), r = 5 Center (0, 0), r = 10 2. x 2 y 2 100 3. x 5 y 4 49 Center (5, -4), r = 7 2 2 4. x 7 y 3 3 Center (-7, 3), r = 3 2 2 5. x y 1 12 Center (0, 1), r = 12 6. x 3 y 2 81 Center (3, 0), r = 9 2 2 2 Writing the Equation of a Circle 1. 2. 3. 4. 5. 6. Center (0, 0) r = 2 x2 + y2 = 4 Center (0, 1) r = 6 x2 + (y – 1)2 = 36 Center (-3, 5) r = 2.5 (x + 3)2 + (y– 5)2= 6.25 2 + (y–10)2= 100 (x + 5) Center (-5, 10) r = 10 2 + y 2= 1 (x – 8) Center (8, 0) r = 1 2 + (y– 9)2= 11.56 (x– 6) Center (6, 9) r = 3.4 Writing the Equation of a circle 2. A circle whose center is at (-3, 2) passes through (-7, 2). a. What is the length of the radius of the circle? b. Write the equation of the circle. Answers: a. r = 4 b. (x + 3)2 + (y - 2)2 = 16 Graphing a Circle Find the x y 9 2 Answers: center (0, 0) radius = 3 2 center and the radius and graph the circle. Graphing a Circle Find the x 1 y 2 2 Answers: center (1, -2) radius = 5 2 25 center and the radius and graph the circle. Graphing a Circle Find the 2 x 3 y 4 2 Answers: center (3, 0) radius = 2 center and the radius and graph the circle. Writing the Equation of a circle 3. A circle has a diameter with endpoints A (1, 2) and B (3, 6). a. What is the center of the circle? The midpoint of segment AB! b. What is the radius of the circle? The distance from the center to A or B! c. What is the equation of the circle? Answers: a. (2, 4) b. sqrt (5) c. (x – 2)2 + (y – 4)2 = 5 Finding the midpoint For the last problem it was necessary to find the midpoint, or the point halfway between two points. There is a formula for this. Part II Midpoint Reminder: What is a Midpoint? • The midpoint of a segment AB is the point that divides AB into two congruent segments. • Where is the midpoint of AB? A Here it is! midpoint B Over Here ? Over Here ? Over Here ? Midpoint on a Number Line • To find the midpoint of two points on a number line, just average the coordinates. ab • Find the midpoint of GT. 2 G T x • Take the average of the coordinates: ab 2 4 9 2 5 2 = 2.5 midpoint Finding a Midpoint in The Coordinate Plane We can find the midpoint between any two points in the coordinate plane by finding the midpoint of the xcoordinates and the midpoint of the y-coordinates. y Example Find the midpoint of the two points. midpoint? x Finding a Midpoint in The Coordinate Plane First: Find the average (midpoint) of the x-coordinates. Remember: Take the average of the two coordinates. a b 4 8 4 2 2 2 2 y a b 2 3 1 0.5 2 2 2 –4 x 2 8 average of x-coordinates Finding a Midpoint in The Coordinate Plane Next: Find the midpoint (average) of the y-coordinates. Remember: Take the average of the two coordinates. a b 2 3 1 0.5 2 2 2 y a b 2 3 1 0.5 2 2 2 3 average of y-coordinates 0.5 x – 22 average of x-coordinates Finding a Midpoint in The Coordinate Plane Finally: The midpoint is the ordered pair: (average of x-coordinates, average of y-coordinates) y = (2, 0.5) a b 2 3 1 0.5 2 2 2 midpoint of y-coordinates 0.5 (2, 0.5) x 2 midpoint of x-coordinates The Midpoint Formula The following formula combines what we did: midpoint = (average of x-coordinates, average of y-coordinates) x1 x2 y1 y2 , 2 2 where (x1, y1) and (x2, y2) are the ordered pairs corresponding to the two points. So let’s go back to the example. Example Find the midpoint of the two points. Solution: We already know the coordinates of the two points. y (8, 3) midpoint? x (– 4, – 2) Example cont. Solution cont. Since the ordered pairs are (x1, y1) = (-4, -2) and (x2, y2) = (8, 3) Plug in x1 = -4, y1 = -2, x2 = 8 and y2 = 3 into midpoint = x1 x2 y1 y2 2 , 2 = 4 8 2 3 2 , 2 = 4 1 2,2 = (2, 0.5) THINK ABOUT IT Find the center, the length of the radius, and write the equation of the circle if the endpoints of a diameter are (-8,2) and (2,0). Center: Use midpoint formula! 8 2 2 0 , 3,1 2 2 Length: use distance formula with center and an endpoint (2 (3)) 2 (0 1) 2 26 Equation: Put it all together x (3) ( y 1) 2 2 26 2 or x 3 2 ( y 1) 26 2