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Chapter 7 Lesson 6. Objective: To find the measures of central angles and arcs and the circumference. Central Angles and Arcs. In a plane, a circle is the set of all points. The set of all points equidistant from a given point is the center .

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Chapter 7 lesson 6

Chapter 7 Lesson 6

Objective: To find the measures of central angles and arcs and the circumference.


Chapter 7 lesson 6

Central Angles

and Arcs

  • In a plane, a circle is the set of all points.

  • The set of all points equidistant from a given point is the center.

  • A radius is a segment that has one endpoint at the center and the other endpoint of the circle.

  • A diameter is a segment that contains the center of a circle and has both endpoints on the circle.


Chapter 7 lesson 6

Congruent Circles have congruent radii.

5 m

5 m

Central Angle is an angle whose vertex is the center of the circle.

A

D

C

B


Chapter 7 lesson 6

Example 1

Finding Central Angles

**Remember a circle measures 360°.**

Sleep: 31% of 360 .31•360=111.6

Food: 9% of 360 .09•360=32.4

Work: 20% of 360 .20•360=72

Must Do: 7% of 360 .07•360=25.2

Entertainment: 18% of 360 .18•360=64.8

Other: 15% of 360 .15•360=54


Chapter 7 lesson 6

  • An arc is a part of a circle.

  • Types of arcs

  • Semicircle is half of a circle.

A

DAE

Minor arc

Major arc

AB

ADB

  • A minor arc is smaller than a semicircle.

  • A major arc is greater than a semicircle.

D


Example 2 identifying arcs

Identify the following in O.  

C

A

O

E

D

Example 2:Identifying Arcs

  • the minor arcs

  • the semicircles

  • 3. the major arcs that contain point A


Chapter 7 lesson 6

Example 3:Identifying Arcs

Identify the minor arcs, major arcs and semicircles in O with point A as an endpoint.  

D

A

  • minor arcs

  • AD, AE

O

B

E

  • major arcs

  • ADE, AED

  • semicircles

  • ADB, AEB


Chapter 7 lesson 6

Adjacent arcs are arcs of the same circle that have exactly one point in common.

Postulate 7-1: Arc Addition Postulate

The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs.

mABC = mAB + mBC

C

B

A


Chapter 7 lesson 6

Example 4:Finding the Measures of Arcs

Find the measure of each arc.

  • BC

58°

D

C

  • BD

B

32°

O

  • ABC

A

  • AB

ABC is a semicircle.


Chapter 7 lesson 6

Example 5:Finding the Measures of Arcs

Find mXY and mDXM in C.

M

mXY = mXD + mDY

mXY = 40 + 56 =

96

Y

W

C

56°

mDXM = mDX + 180

D

40°

mDXM = 40 + 180

X

mDXM = 220


Chapter 7 lesson 6

The circumference of a circle is the distance around the circle.

The number pi (π) is the ratio of the circumference of a circle to its diameter.

Theorem 7-13Circumference of a Circle

The circumference of a circle is π times the diameter.



Chapter 7 lesson 6

Example 6: are

Concentric Circles

A car has a turning radius of 16.1 ft. The distance between the two front tires is 4.7 ft. In completing the (outer) turning circle, how much farther does a tire travel than a tire on the concentric inner circle?

circumference of outer circle = C = 2πr = 2π(16.1) = 32.2π

To find the radius of the inner circle, subtract 4.7 ft from the turning radius.

radius of the inner circle = 16.1 − 4.7 = 11.4

circumference of inner circle = C = 2πr = 2π(11.4) = 22.8π

The difference in the two distances is 32.2π − 22.8π, or 9.4π.

A tire on the turning circle travels about 29.5 ft farther than a tire on the inner circle.


Chapter 7 lesson 6

The measure of an arc is in degrees while the are arc length is a fraction of a circle's circumference.

Theorem 7-14Arc Length

The length of an arc of a circle is the product of the ratio                  and the circumference of the circle.

length of    =      • 2πr


Example 7 finding arc length
Example 7: are Finding Arc Length

Find the length of each arc shown in red. Leave your answer in terms of π.


Example 8 finding arc length
Example 8: are Finding Arc Length

Find the length of a semicircle with radius of 1.3m. Leave your answer in terms of π.


Chapter 7 lesson 6

B are

18 cm

150°

M

D

A

Example 9:Finding Arc Length

Find he length of ADB in terms of π.


Chapter 7 lesson 6

Congruent arcs are are arcs that have the same measure and are in the same circle or in congruent circles.


Assignment
Assignment: are

Pages 389-392

#1-39