10.7 write and graph equations of circles - rjs · pdf filewrite and graph equations of...

Your Notes

10.7

Copyright © Holt McDougal. All rights reserved. Lesson 10.7 • Geometry Notetaking Guide 289

VOCABULARY

Standard equation of a circle

Write the equation of the circle shown.

x

y

1

1

Solution

The radius is and the center is at .

x2 1 y2 5 2 Equation of circle

x2 1 y2 5 2 Substitute.

x2 1 y2 5 Simplify.

The equation of the circle is x2 1 y2 5 .

Example 1 Write an equation of a circle

1. Write an equation of the

x

y

2

2

circle shown.

Checkpoint Complete the following exercise.

Write and Graph Equations of CirclesGoal p Write equations of circles in the coordinate plane.

Your Notes

10.7


VOCABULARY

Standard equation of a circle The standard equation of a circle with center (h, k) and radius r is (x 2 h)2 1 (y 2 k)2 5 r2.

Write the equation of the circle shown.

x

y

1

1

Solution

The radius is 2 and the center is at the origin .

x2 1 y2 5 r 2 Equation of circle

x2 1 y2 5 2 2 Substitute.

x2 1 y2 5 4 Simplify.

The equation of the circle is x2 1 y2 5 4 .

Example 1 Write an equation of a circle

1. Write an equation of the

x

y

2

2

circle shown.

x2 1 y2 5 36


Write and Graph Equations of CirclesGoal p Write equations of circles in the coordinate plane.

LAH_GE_11_FL_NTG_Ch10_267-294.indd 289LAH_GE_11_FL_NTG_Ch10_267-294.indd 289 1/13/09 3:40:18 PM1/13/09 3:40:18 PM

Your NotesSTANDARD EQUATION OF A CIRCLE

The standard equation of a circle with center (h, k) and radius r is:

(x 2 h)2 1 (y 2 k)2 5 r2

Write the standard equation of a circle with center (0, 25) and radius 3.7.

(x 2 h)2 1 (y 2 k)2 5 r2 Standard equation of a circle

(x 2 )2 1 (y 2 ( ))2 5 2 Substitute.

x2 1 (y 1 )2 5 Simplify.

Example 2 Write the standard equation of a circle

The point (23, 4) is on a circle with

x

y

1

1

(21, 2)

(23, 4)center (21, 2). Write the standard equation of the circle.

SolutionTo write the standard equation, you need to know the values of h, k, and r. To find r, find the distance between the and the point (23, 4) on the circle.

r 5 Ï}}}

[23 2 ( )]2 1 ( 2 2)2 Distance formula

5 Ï}}

( )2 1 2 Simplify.

5 Simplify.

Substitute (h, k) 5 (21, 2) and r 5 into the standard equation of a circle.


(x 2 ( ))2 1 (y 2 )2 5 ( )2 Substitute.

(x 1 )2 1 (y 2 )2 5 Simplify.

The standard equation of the circle is (x 1 )2 1 (y 2 )2 5 .


290 Lesson 10.7 • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Your NotesSTANDARD EQUATION OF A CIRCLE

The standard equation of a circle with center (h, k) and radius r is:

(x 2 h)2 1 (y 2 k)2 5 r2

Write the standard equation of a circle with center (0, 25) and radius 3.7.


(x 2 0 )2 1 (y 2 ( 25 ))2 5 3.7 2 Substitute.

x2 1 (y 1 5 )2 5 13.69 Simplify.


The point (23, 4) is on a circle with

x

y

1

1

(21, 2)

(23, 4)center (21, 2). Write the standard equation of the circle.

SolutionTo write the standard equation, you need to know the values of h, k, and r. To find r, find the distance between the center and the point (23, 4) on the circle.

r 5 Ï}}}

[23 2 ( 21 )]2 1 ( 4 2 2)2 Distance formula

5 Ï}}

( 22 )2 1 2 2 Simplify.

5 2 Ï}

2 Simplify.

Substitute (h, k) 5 (21, 2) and r 5 2 Ï}

2 into the standard equation of a circle.


(x 2 ( 21 ))2 1 (y 2 2 )2 5 ( 2 Ï}

2 )2 Substitute.

(x 1 1 )2 1 (y 2 2 )2 5 8 Simplify.

The standard equation of the circle is (x 1 1 )2 1 (y 2 2 )2 5 8 .




Your Notes


The equation of a circle is (x 2 2)2 1 (y 1 3)2 5 16. Find the center and radius. Graph the circle.

Solution

x

y2

2Rewrite the equation to find the center and radius.

(x 2 2)2 1 (y 1 3)2 5 16

(x 2 2)2 1 [y 2 ( )]2 5

The center is ( , ) and the radius is . Use a compass to graph the circle.

Example 4 Graph a circle

If you know the equation of a circle, you can graph the circle by identifying its center and radius.

2. Write the standard equation of a circle with center (23, 25) and radius 6.1.

3. The point (21, 2) is on a circle with center (3, 23). Write the standard equation of the circle.

4. The equation of a circle is (x 1 2)2 1 (y 2 1)2 5 9. Graph the circle.

x

y

1

1

Checkpoint Complete the following exercises.

Your Notes


The equation of a circle is (x 2 2)2 1 (y 1 3)2 5 16. Find the center and radius. Graph the circle.

Solution

x

y2

2

(2, 23)

Rewrite the equation to find the center and radius.

(x 2 2)2 1 (y 1 3)2 5 16

(x 2 2)2 1 [y 2 ( 23 )]2 5 42

The center is ( 2 , 23 ) and the radius is 4 . Use a compass to graph the circle.

Example 4 Graph a circle

If you know the equation of a circle, you can graph the circle by identifying its center and radius.

2. Write the standard equation of a circle with center (23, 25) and radius 6.1.

(x 1 3)2 1 (y 1 5)2 5 37.21

3. The point (21, 2) is on a circle with center (3, 23). Write the standard equation of the circle.

(x 2 3)2 1 (y 1 3)2 5 41

4. The equation of a circle is (x 1 2)2 1 (y 2 1)2 5 9. Graph the circle.

x

y

1

1

(22, 1)

Checkpoint Complete the following exercises.


Homework

Your Notes

Time Capsule You bury a time capsule and use a grid to write directions for finding it. Use the following measurements to find the burial location of the time capsule.

• The capsule is about 11 feet from the oak tree at A(0, 0).

• The capsule is 8 feet from the flagpole at B(0, 8).

• The capsule is 4 feet from the mailbox at C(212, 8).

Solution

x

y

4

4

The set of all points equidistant from a given point is a circle, so the burial location is located on each of the following circles.

(A with center ( , ) and radius

(B with center ( , ) and radius

(C with center ( , ) and radius

To find the burial location, graph the circles on a graph where units are measured in feet. Estimate the point of of all three circles.

The burial location is at about ( , ).

Example 5 Use graphs of circles

5. In Example 4, suppose the mailbox is at C(12, 8) and the time capsule is 4 feet away. Find the burial location of the time capsule.



Homework

Your Notes

Time Capsule You bury a time capsule and use a grid to write directions for finding it. Use the following measurements to find the burial location of the time capsule.

• The capsule is about 11 feet from the oak tree at A(0, 0).

• The capsule is 8 feet from the flagpole at B(0, 8).

• The capsule is 4 feet from the mailbox at C(212, 8).

Solution

x

y

4

4A

BCThe set of all points equidistant

from a given point is a circle, so the burial location is located on each of the following circles.

(A with center ( 0 , 0 ) and radius 11

(B with center ( 0 , 8 ) and radius 8

(C with center ( 212 , 8 ) and radius 4

To find the burial location, graph the circles on a graph where units are measured in feet. Estimate the point of intersection of all three circles.

The burial location is at about ( 28 , 8 ).

Example 5 Use graphs of circles

5. In Example 4, suppose the mailbox is at C(12, 8) and the time capsule is 4 feet away. Find the burial location of the time capsule.

(8, 8)




Words to ReviewGive an example of the vocabulary word.

Circle

Chord

Tangent

Central angle

Semicircle

Center, radius, diameter of a circle

Secant

Concentric Circles

Minor arc, Major arc

Measure of a minor arc, major arc

Copyright © Holt McDougal. All rights reserved. Words to Review • Geometry Notetaking Guide 293

LAH_GE_11_FL_NTG_Ch10_267-294.indd 293LAH_GE_11_FL_NTG_Ch10_267-294.indd 293 12/17/09 1:51:47 AM12/17/09 1:51:47 AM

Words to ReviewGive an example of the vocabulary word.

Circle

Chord

chord

Tangent

tangent

Central angle

centralangle

Semicircle

A

B

PC

C ABC is a semicircle.

Center, radius, diameter of a circlediameter

center radius

Secant

secant

Concentric Circles

Minor arc, Major arcA

BP

C

C AB is a minor arc.

C ACB is a major arc.

Measure of a minor arc, major arc

A

BP

C

928

m C AB 5 928, m C ACB 5 2688

Copyright © Holt McDougal. All rights reserved. Words to Review • Geometry Notetaking Guide 293

LAH_GE_11_FL_NTG_Ch10_267-294.indd 293LAH_GE_11_FL_NTG_Ch10_267-294.indd 293 12/17/09 1:51:47 AM12/17/09 1:51:47 AM

Congruent circles

Inscribed angle

Inscribed polygon

Segments of a chord

External segment

Congruent arcs

Intercepted arc

Circumscribed circle

Secant segment


Review your notes and Chapter 10 by using the Chapter Review on pages 735–738 of your textbook.

294 Words to Review • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Congruent circles

3

3

Inscribed angle

inscribedangle

Inscribed polygon

inscribedpolygon

Segments of a chord

segmentsof a chord

External segment

externalsegment

Congruent arcs

908908

A

B

C

C AB

>

C BC

Intercepted arc

interceptedarc

Circumscribed circle

circumscribedcircle

Secant segment

secantsegment


(x 2 h)2 1 (y 2 k)2 5 r2 where the center of the circle is (h, k) and the radius is r.

Review your notes and Chapter 10 by using the Chapter Review on pages 735–738 of your textbook.

294 Words to Review • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved.


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