1. 10.6 Equations of Circles
Unit IIIF Day 3

2. Do Now
What is the distance formula for two points, (x1, y1) and (x2, y2)?
What is the definition of a circle?

3. Quick Recap: Angle Measures
List the three angle formulas from this chapter (10.4)
On:
Inside:
Outside:
angle = ½ arc
angle = ½ sum of arcs (aka average)
angle = ½ difference of arcs

4. Quick Recap: Segment Lengths
List the three segment formulas from this chapter in your own words (10.5)
two chords:
two secant segments:
secant segment and tangent segment:
product of one chord’s segments = product of other chord’s segments
whole secant segment * external segment = other whole secant segment * its external segment
(tangent segment)2 = whole secant segment * external segment

5. Finding Equations of Circles
Let radius = r and the center be (h, k); Let (x, y) be any point on the circle. Find the equation for a circle.
You can write an equation of a circle in a coordinate plane if you know its radius and the coordinates of its center.
Suppose the radius is r and the center is (h, k). Let (x, y) be any point on the circle. The distance between (x, y) and (h, k) is r, so you can use the Distance Formula: r = √[(x – h)2 + (y – k)2]
Square both sides to find the standard equation of a circle with radius r and center (h, k).
(x – h)2 + (y – k)2 = r2
If the center is at the origin, then the standard equation is
x2 + y2 = r2.

6. Ex. 1: Writing a Standard Equation of a Circle
Write the standard equation of the circle with a center at (-4, 0) and radius 7.1 units.
(x – h)2 + (y – k)2 = r2
[(x – (-4)]2 + (y – 0)2 = 7.12
(x + 4)2 + (y – 0)2 = 50.41

7. Ex. 2: Writing a Standard Equation of a Circle
The point (1, 2) is on a circle whose center is (5, -1). Find the radius of the circle. Then write the equation of the circle in standard form.
r2 = (5 – 1)2 + (-1 – 2)2 ) = 25
r = 5
So 25 = (x – 5)2 + (y + 1)2

8. Ex. 3: Graphing a circle
The equation of a circle is (x + 2)2 + (y – 3)2 = 9. Graph the circle.
If you know the equation of a circle, you can graph the circle by identifying its center and radius.
First rewrite the equation to find the center and its radius.
(x+2)2 + (y-3)2 = 9
[x – (-2)]2 + (y – 3)2=32
The center is (-2, 3) and the radius is 3.
To graph the circle, place the point of a compass at (-2, 3), set the radius at 3 units, and swing the compass to draw a full circle.

9. Ex. 4: Applying Graphs of Circles
A theater light illuminates a circular area on the stage. The equation (x – 13)2 + (y – 4)2 = 16 represents a disk of light.
Graph the disk of light.
Henry is at (11, 4), Jolene is at (8, 5), and Mark is at (15, 5). Which actors are in the disk of light?
Rewrite the equation to find the center and radius.
(x – h)2 + (y – k)2= r2
(x – 13)2 + (y – 4)2 = 16
(x – 13)2 + (y – 4)2= 42
The center is at (13, 4) and the radius is 4.
(x - 13)2 + (y - 4)2 = 16
The graph shows that Henry and Mark are both in the disk of light.

10. Closure
The point (0, –1) is on a circle whose center is (–4, 1). Write the standard equation of the circle.
(x + 4)2+ (y – 1)2 = 20