1. LESSON 10-4 Equations of Circles
Created by Lisa Palen and Kristina Green
Henrico High School

2. Part I
Equations of Circles

3. 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

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

5. 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

6. 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

7. 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

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

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

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

11. 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

12. 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.

13. Part II
Midpoint

14. 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?

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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.

21. 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)

22. 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)

23. 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