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### Chapter 7 Lesson 6

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

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.

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

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

- 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

C

A

•

O

E

D

Example 2:Identifying Arcs- the minor arcs
- the semicircles
- 3. the major arcs that contain point A

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

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

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.

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

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.

Circles that lie in the same plane and have the same center are concentric circles.

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.

The measure of an arc is in degrees while the 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

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

Example 8:Finding Arc Length

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

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

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