Lesson 11

A New Way to Measure Angles

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Problem 1

Here is a central angle that measures 1.5 radians. Select all true statements.

Circle with central angle of 1 point 5 radians.
A:

The radius is 1.5 times longer than the length of the arc defined by the angle.

B:

The length of the arc defined by the angle is 1.5 times longer than the radius.

C:

The ratio of arc length to radius is 1.5.

D:

The ratio of radius to arc length is 1.5.

E:

The area of the whole circle is 1.5 times the area of the slice.

F:

The circumference of the whole circle is 1.5 times the length of the arc formed by the angle.

Solution

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Problem 2

Match each arc length \(\ell\) and radius \(r\) with the measure of the central angle of the arc in radians.

A:

\(r=2, \ell=\frac{\pi}{2}\)

B:

\(r=3, \ell=2\)

C:

\(r=3.5, \ell=2.8\)

D:

\(r=4, \ell=3\)

E:

\(r=5, \ell=\pi\)

F:

\(r=6, \ell=4\)

1:

\(\frac{\pi}{5}\) radians

2:

\(\frac{2}{3}\) radians

3:

0.75 radians

4:

\(\frac{\pi}{4}\) radians

5:

0.8 radians

Solution

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Problem 3

Han thinks that since the arc length in circle A is longer, its central angle is larger. Do you agree with Han? Show or explain your reasoning.

circle A

A circle with sector with a central angle. Radius 15. Arc measure 6 pi.

circle B

Circle with radius 5 units. Arch length 2 pi drawn.

Solution

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Problem 4

Circle B is a dilation of circle A.

circle A

circle, radius = 4 units. 15 degree sector on circle.

circle B

circle, radius = 2 units. 15 degree sector on circle.
  1. What is the scale factor?
  2. What is the area of the 15 degree sector in circle A?
  3. What is the area of the 15 degree sector in circle B?
  4. What is the ratio of the areas of the sectors?
  5. How does the ratio of areas of the sectors compare to the scale factor?

Solution

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(From Unit 7, Lesson 10.)

Problem 5

Priya and Noah are riding different size Ferris wheels at a carnival. They started at the same time. The highlighted arcs show how far they have traveled.

Noah’s Ferris wheel

A circle with sector with a central angle labeled 135 degrees, radius 15 meters. Arc measure in blue is unknown.

Priya’s Ferris wheel

A circle with sector with a central angle labeled 135 degrees. Radius 30 meters. Unknown arc measure in blue.
  1. How far has Noah traveled?
  2. How far has Priya traveled?
  3. If the Ferris wheels will each complete 1 revolution, who do you think will finish first?

Solution

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(From Unit 7, Lesson 10.)

Problem 6

A circle has radius 8 units, and a central angle is drawn in. The length of the arc defined by the central angle is \(4\pi\) units. Find the area of the sector outlined by this arc.

Solution

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(From Unit 7, Lesson 9.)

Problem 7

Clare is trying to explain how to find the area of a sector of a circle. She says, “First, you find the area of the whole circle. Then, you divide by the radius.“ Do you agree with Clare? Explain or show your reasoning.

Solution

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(From Unit 7, Lesson 8.)

Problem 8

Line \(BD\) is tangent to a circle with diameter \(AB\). List 2 right angles.

Triangle A B C is inscribed in a circle with A B as the diameter and all points on the circle. Line B D is tangent to the circle.

Solution

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(From Unit 7, Lesson 3.)