Lesson 12

Radian Sense

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Problem 1

Match each diagram showing a sector with the measure of its central angle in radians.

Circles A through F with sectors shown.
A:

diagram A

B:

diagram B

C:

diagram C

D:

diagram D

E:

diagram E

F:

diagram F

1:

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

2:

\(\pi\)

3:

\(\frac{\pi}{3}\)

4:

\(\frac{\pi}{2}\)

5:

\(2\pi\)

6:

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

Solution

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Problem 2

In the circle, sketch a central angle that measures \(\frac{2\pi}{3}\) radians.

Circle with center point.

Solution

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Problem 3

Angle \(AOC\) has a measure of \(\frac{5\pi}{6}\) radians. The length of arc \(AB\) is \(2\pi\) units and the radius is 12 units. What is the area of sector \(BOC\)?

Angle A O C. Point B falls on perimeter, between points A and C. A B labeled 2 pi. A O labeled 12. Sector B O C shaded.

 

Solution

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Problem 4

Calculate the radian measure of a 30 degree angle. Use any method you like, including sketching in the circle diagram provided. Explain or show your reasoning.

Circle with radius 1 unit.

Solution

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(From Unit 7, Lesson 11.)

Problem 5

Lin thinks that the central angle in circle A is congruent to the central angle in circle B. Do you agree with Lin? Show or explain your reasoning.

circle A

Circle with sector with central angle. Radius 12. Arc measure 4 pi.

circle B

Circle with radius = 3. Sector with length = pi.

Solution

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(From Unit 7, Lesson 11.)

Problem 6

circle A

Circle radius of 5 units. A 60 degree sector is noted.

circle B

A circle, radius 9 units. A 90 degree sector is noted.

Select all true statements.

A:

The sector in circle B has a larger area than the sector in circle A.

B:

Not taking into account the sectors, circle A and circle B are similar.

C:

The fraction of the circumference taken up by the arc outlining circle A’s sector is smaller than the fraction of the circumference taken up by the arc in circle B.

D:

The ratio of the area of circle A’s sector to its total area is \(\frac16\).

E:

The ratio of circle A’s area to circle B’s area is \(\frac 59\).

Solution

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(From Unit 7, Lesson 10.)

Problem 7

Match each arc length and radius with the measure of the central angle that defines the arc.

A:

arc length \(5\pi\) cm, radius 5 cm

B:

arc length \(4\pi\) cm, radius 10 cm

C:

arc length \(9\pi\) cm, radius 12 cm

D:

arc length \(3\pi\) cm, radius 18 cm

1:

angle: 30 degrees

2:

angle: 72 degrees

3:

angle: 135 degrees

4:

angle: 180 degrees

Solution

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(From Unit 7, Lesson 9.)

Problem 8

Quadrilateral \(ABCD\) is shown with the given angle measures. Select all true statements.

Quadrilateral A B C D inscribed in a circle. Angle A B C is 125 degrees, angle B C D is 40 degrees.
A:

Angle \(A\) measures 140 degrees.

B:

The measures of angle \(A\) and angle \(D\) must add to 180 degrees.

C:

Angle \(A\) measures 55 degrees.

D:

Angle \(D\) measures 55 degrees.

E:

Angle \(D\) measures 40 degrees.

Solution

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(From Unit 7, Lesson 4.)