formulas things you should know at this point. measure of an inscribed angle

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


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


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


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


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


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.


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.


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.


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.


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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formulas things you should know at this point. measure of an inscribed angle

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