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FormulasThings you should know at this point
Measure of an Inscribed Angle
• If an angle is inscribed in a circle, then its measure is one half the measure of its intercepted arc.
= AB
C
B
A
D
Theorem. The measure of an formed by 2 lines that intersect inside a circle is
𝑚1=12(𝑥+ 𝑦 )
Measure of intercepted arcs𝟏 𝒙 °
𝒚 °
Theorem. The measure of an formed by 2 lines that intersect outside a circle is
𝒎𝟏=𝟏𝟐(𝒙−𝒚 )
Smaller Arc
Larger Arc
x°
y°
1
x°
y°
1
2 Secants:
x°
y°
1
Tangent & a Secant 2 Tangents
3 cases:
Lengths of Secants, Tangents, & Chords
2 Chords
𝑎•𝑏=𝑐 •𝑑
2 Secants
𝑎 𝑐𝑏
𝑑 𝑥
𝑤
𝑧
𝑦
𝑤(𝑤+𝑥)=𝑦 (𝑦+𝑧)
Tangent & Secant
𝑡𝑦
𝑧
𝑡 2=𝑦 (𝑦+𝑧 )
12-5 Circles in the Coordinate Plane
Bonus: Completing the Square
Write equations and graph circles in the coordinate plane.
Use the equation and graph of a circle to solve problems.
Objectives
The equation of a circle is based on the Distance Formula and the fact that all points on a circle are equidistant from the center.
Example 1A: Writing the Equation of a Circle
Write the equation of each circle.
J with center J (2, 2) and radius 4
(x – h)2 + (y – k)2 = r2
(x – 2)2 + (y – 2)2 = 42
(x – 2)2 + (y – 2)2 = 16
Equation of a circle
Substitute 2 for h, 2 for k, and 4 for r.
Simplify.
Example 1B: Writing the Equation of a Circle
Write the equation of each circle.
K that passes through J(6, 4) and has center K(1, –8)
Distance formula.
Simplify.
(x – 1)2 + (y – (–8))2 = 132
(x – 1)2 + (y + 8)2 = 169
Substitute 1 for h, –8 for k, and 13 for r.Simplify.
Check It Out! Example 1a
Write the equation of each circle.
P with center P(0, –3) and radius 8
(x – h)2 + (y – k)2 = r2
(x – 0)2 + (y – (–3))2 = 82
x2 + (y + 3)2 = 64
Equation of a circle
Substitute 0 for h, –3 for k, and 8 for r.
Simplify.
Check It Out! Example 1b
Write the equation of each circle.
Q that passes through (2, 3) and has center Q(2, –1)
Distance formula.
Simplify.
(x – 2)2 + (y – (–1))2 = 42
(x – 2)2 + (y + 1)2 = 16
Substitute 2 for h, –1 for k, and 4 for r.Simplify.
If you are given the equation of a circle, you can graph the circle by making a table or by identifying its center and radius.
Example 2A: Graphing a Circle
Graph x2 + y2 = 16.
Step 1 Make a table of values.Since the radius is , or 4, use ±4 and use the values between for x-values.
Step 2 Plot the points and connect them to form a circle.
Example 2B: Graphing a Circle
Graph (x – 3)2 + (y + 4)2 = 9.
The equation of the given circle can be written as (x – 3)2 + (y – (– 4))2 = 32.
So h = 3, k = –4, and r = 3.
The center is (3, –4) and the radius is 3. Plot the point (3, –4). Then graph a circle having this center and radius 3.
(3, –4)
Check It Out! Example 2a
Graph x² + y² = 9.
Step 2 Plot the points and connect them to form a circle.
Since the radius is , or 3, use ±3 and use the values between for x-values.
x 3 2 1 0 –1 –2 –3
y 0 2.2 2.8 3 2.8 2.2 0
Check It Out! Example 2b
Graph (x – 3)2 + (y + 2)2 = 4.
The equation of the given circle can be written as (x – 3)2 + (y – (– 2))2 = 22.
So h = 3, k = –2, and r = 2.
The center is (3, –2) and the radius is 2. Plot the point (3, –2). Then graph a circle having this center and radius 2.
(3, –2)
Lesson Quiz: Part I
Write the equation of each circle.
1. L with center L (–5, –6) and radius 9
(x + 5)2 + (y + 6)2 = 81
2. D that passes through (–2, –1) and has center D(2, –4)
(x – 2)2 + (y + 4)2 = 25
Lesson Quiz: Part II
Graph each equation.
3. x2 + y2 = 4
4. (x – 2)2 + (y + 4)2 = 16
Review
Standard Equation of a Circle
9.3 Circles
center:
radius:
Review
-coordinate of a point on a circle
-coordinate of a point on a circle
Radius
-coordinate of the center of the circle
-coordinate of the center of the circle
Completing the SquareRecall that we can solve quadratic equations of the form by “completing the square”…
1. Subtract c from both sides.
2. Square half the coefficient of x and add it to both sides.
3. Then “un-FOIL”
We can use this process to find the standard equation of circles when given the “FOILed” equation.
2
2
b
2
2
b
Completing the Square to Find the Equation of a Circle
Find the center and radius of the circle
.
Find the center and radius of a circle
Example 3 FINDING THE CENTER AND RADIUS BY COMPLETING THE SQUARE
Show that x2 – 6x + y2 +10y + 25 = 0 has a circle as a graph. Find the center and radius.
Solution We complete the square twice, once for x and once for y.
2 26 10 25 0x x y y 2 2( ) ( )6 10 25x x y y
221
( ) (6 93)2
and2
21( ) 50 251
2
Example 3 FINDING THE CENTER AND RADIUS BY COMPLETING THE SQUARE
Add 9 and 25 on the left to complete the two squares, and to compensate, add 9 and 25 on the right.
2 2( 6 ) ( 109 25) 9 2525x x y y
Add 9 and 25 on both
sides.
2 2( 3) ( 5) 9x y Factor
Complete the square.
Since 9 > 0, the equation represents a circle with center at (3, – 5) and radius 3.
Example 4 FINDING THE CENTER AND RADIUS BY COMPLETING THE SQUARE
Show that 2x2 + 2y2 – 6x +10y = 1 has a circle as a graph. Find the center and radius.
Solution To complete the square, the coefficients of the x2- and y2-terms must be 1.
2 2 6 12 02 1x y x y
2 22 3 2 5 1x x y y Group the terms; factor out 2.
Example 4 FINDING THE CENTER AND RADIUS BY COMPLETING THE SQUARE
2 22 3 2 5 1x x y y Group the terms; factor out 2.
2 2 252
4
2
92
4
92
4
5
24
3
15
x x y y
Be careful here.
Example 4 FINDING THE CENTER AND RADIUS BY COMPLETING THE SQUARE
Factor; simplify on the right.
2 2 252
4
2
92
4
92
4
5
24
3
15
x x y y
2 23 5
2 2 182 2
x y
2 23 5
92 2
x y
Divide both sides by 2.
Example 4 FINDING THE CENTER AND RADIUS BY COMPLETING THE SQUARE
2 23 5
92 2
x y
Divide both sides by 2.
3 5The equation has a circle with center at ,
2 2
and radius 3 as its graph.
Deriving The Quadratic Formula
2 0b c
x xa a
Divide both sides by a
2 22
2 2
b b c bx x
a a a a
2 2
2 2
4
2 4 4
b b acx
a a a
2
2
4
2 4
b b acx
a a
2 4
2
b b acx
a
Complete the square by adding (b/2a)2 to both sides
Factor (left) and find LCD (right)
Combine fractions and take the square root of both sides
Subtract b/2a and simplify
2If 0 (and 0 then:),ax bx c a
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