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

Equation of a Circle Center (0, 0) Radius = r Center (h, k) Radius = r

Finding the Center and the Radius when given the equation Center (0, 0), r = 5 Center (0, 0), r = 10 Center (5, -4), r = 7 Center (-7, 3), r = Center (0, 1), r = Center (3, 0), r = 9

Writing the Equation of a Circle Center (0, 0) r = 2 Center (0, 1) r = 6 Center (-3, 5) r = 2.5 Center (-5, 10) r = 10 Center (8, 0) r = 1 Center (6, 9) r = 3.4 x2 + y2 = 4 x2 + (y – 1)2 = 36 (x + 3)2 + (y– 5)2= 6.25 (x + 5)2 + (y–10)2= 100 (x – 8)2 + y2= 1 (x– 6)2 + (y– 9)2= 11.56

Writing the Equation of a circle 2. A circle whose center is at (-3, 2) passes through (-7, 2). What is the length of the radius of the circle? Write the equation of the circle. Answers: a. r = 4 b. (x + 3)2 + (y - 2)2 = 16

Graphing a Circle Find the center and the radius and graph the circle. Answers: center (0, 0) radius = 3

Graphing a Circle Find the center and the radius and graph the circle. Answers: center (1, -2) radius = 5

Graphing a Circle Find the center and the radius and graph the circle. Answers: center (3, 0) radius = 2

Writing the Equation of a circle 3. A circle has a diameter with endpoints A (1, 2) and B (3, 6). What is the center of the circle? What is the radius of the circle? What is the equation of the circle? The midpoint of segment AB! diameter The distance from the center to A or B! 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? Here it is! A Over Here? midpoint B Over Here? Over Here?

Midpoint on a Number Line To find the midpoint of two points on a number line, just average the coordinates. Find the midpoint of GT. G T x Take the average of the coordinates: midpoint = 2.5

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 x-coordinates 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. y – 4 x 8 2 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. y 3 0.5 average of y-coordinates x – 2 2 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) = (2, 0.5) y (2, 0.5) 0.5 midpoint of y-coordinates x 2 midpoint of x-coordinates

The Midpoint Formula midpoint The following formula combines what we did: midpoint = (average of x-coordinates, average of y-coordinates) where (x1, y1) and (x2, y2) are the ordered pairs corresponding to the two points. So let’s go back to the example.

Example (8, 3) (– 4, – 2) 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 = = = (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! Length: use distance formula with center and an endpoint Equation: Put it all together