A New Way to Measure Angles

ID: pipud-kohot
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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Geometry
Grade: 9-12
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21 questions

A New Way to Measure Angles

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Problem 1

1) Below is a central angle that measures 1.5 radians. Select all\textbf{all} true statements. Write each corresponding letter in the answer box and separate letters with commas.

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.

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Problem 2

Match each arc length \ell and radius rr with the measure of the central angle of the arc in radians. Write the numbers of the corresponding central angles in the answer boxes below.

ID number of central anglecentral angle1π5 radians223 radians30.75 radians4π4 radians50.8 radians\begin{array}{|c|c|c|} \hline \\[-1em] \textbf{ID number of central angle} & \textbf{central angle} \\[-1em] \\ \hline \\[-1em] 1 & \frac{\pi}{5} \text{ radians} \\[-1em] \\ \hline \\[-1em] 2 & \frac{2}{3} \text{ radians} \\[-1em] \\ \hline \\[-1em] 3 & 0.75 \text{ radians} \\[-1em] \\ \hline \\[-1em] 4 & \frac{\pi}{4} \text{ radians} \\[-1em] \\ \hline \\[-1em] 5 & 0.8 \text{ radians} \\[-1em] \\ \hline \end{array}

2) r=2, =π2r = 2, \ \ell = \frac{\pi}{2}

3) r=3, =2r = 3, \ \ell = 2

4) r=3.5, =2.8r = 3.5, \ \ell = 2.8

5) r=4, =3r = 4, \ \ell = 3

6) r=5, =πr = 5, \ \ell = \pi

7) r=6, =4r = 6, \ \ell = 4

Problem 3

Han thinks that since the arc length in circle A is longer, its central angle is larger.

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8) Do you agree with Han?

True or false? Write below.

9) Show or explain your answer.

Problem 4

Circle B is a dilation of circle A.

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10) What is the scale factor?

11) What is the area of the 15-degree sector in circle A?

12) What is the area of the 15-degree sector in circle B?

13) What is the ratio of the small circle's area to the large circle's area?

14) How does the ratio of areas of the sectors compare to the scale factor?

a)

The ratio of the area of the sectors is equal to the scale factor.

b)

The ratio of the area of the sectors is three times the scale factor.

c)

The ratio of the area of the sectors is the square of the scale factor.

d)

The ratio of the area of the sectors is four times the scale factor.

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.

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15) How far has Noah traveled? Round your answer to the nearest tenth of a meter.

16) How far has Priya traveled. Round your answer to the nearest tenth of a meter.

17) If the ferris wheels each complete 1 revolution, who will finish first? Assume the ferris wheels each continue at their current paces.

a)

Noah will finish first.

b)

Priya will finish first.

c)

They will finish at the same time.

d)

There is not enough information

to answer the question.

Problem 6

18) A circle has radius 8 units, and a central angle is drawn in. The length of the arc defined by the central angle is 4π4\pi units. Find the area of the sector outlined by this arc.

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.“

19) Do you agree with Clare?

True or false? Write below.

20) Explain or show your reasoning.

Problem 8

21) Line BDBD is tangent to a circle with diameter ABAB. Select all\textbf{all} angles that are right angles. Write each corresponding letter in the answer box and separate letters with commas.

a) Angle ABCABC \quad\quad b) Angle ACBACB\quad\quad c) Angle BACBAC\quad\quad d) AngleDBADBA\quad\quad e) Angle BDABDA\quad\quad f) Angle DABDAB\quad\quad

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