Angles, Arcs, and Radii

ID: pokiv-rapos
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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Geometry
Grade: 9-12
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19 questions

Angles, Arcs, and Radii

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Problem 1

Here are 2 circles. The smaller circle has radius rr, circumference cc, and diameter dd. The larger circle has radius RR, circumference CC, and diameter DD. The larger circle is a dilation of the smaller circle by a factor of kk.

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1) Select all\textbf{all} of the ratios that are equal to kk? Write each corresponding letter in the answer box and separate letters with commas.

a) Cc\frac{C}{c} \quad\quad b) CD\frac{C}{D} \quad\quad c) CR\frac{C}{R} \quad\quad d) Dd\frac{D}{d} \quad\quad e) DR\frac{D}{R} \quad\quad f) Rr\frac{R}{r} \quad\quad g) cd\frac{c}{d} \quad\quad h) dr\frac{d}{r}

2) Select all\textbf{all} of the ratios that are equal to π\pi. Write each corresponding letter in the answer box and separate letters with commas.

a) Cc\frac{C}{c} \quad\quad b) CD\frac{C}{D} \quad\quad c) CR\frac{C}{R} \quad\quad d) Dd\frac{D}{d} \quad\quad e) DR\frac{D}{R} \quad\quad f) Rr\frac{R}{r} \quad\quad g) cd\frac{c}{d} \quad\quad h) dr\frac{d}{r}

3) Select all\textbf{all} of the ratios that are equal to 2. Write each corresponding letter in the answer box and separate letters with commas.

a) Cc\frac{C}{c} \quad\quad b) CD\frac{C}{D} \quad\quad c) CR\frac{C}{R} \quad\quad d) Dd\frac{D}{d} \quad\quad e) DR\frac{D}{R} \quad\quad f) Rr\frac{R}{r} \quad\quad g) cd\frac{c}{d} \quad\quad h) dr\frac{d}{r}

Problem 2

4) Tyler is confident that all circles are similar, but he cannot explain why this is true. Help Tyler explain why all circles are similar.

Problem 3

Circle B is a dilation of circle A.

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5) What is the scale factor of dilation?

6) What is the length of the highlighted arc in circle A? Write your answer in terms of π\pi.

7) What is the length of the highlighted arc in circle B? Write your answer in terms of π\pi.

8) What is the ratio of the arc lengths?

9) How does the ratio of arc length compare to the scale factor?

a)

The ratio of arc length is greater than the scale factor.

b)

The ratio of arc length is less than the scale factor.

c)

They are equal.

Problem 4

Kiran cuts out a square piece of paper with side length 6 inches. Mai cuts out a paper sector of a circle with radius 6 inches, and calculates the arc length to be 4π4\pi inches.

10) Whose paper is larger?

a)

Kiran’s paper is larger.

b)

Mai’s paper is larger.

c)

The two papers are equal in size.

d)

There is not enough information to answer the question.

11) Explain or show your reasoning.

Problem 5

12) A circle has radius 3 centimeters. Suppose an arc on the circle has length 4π4\pi centimeters. What is the measure of the central angle whose radii define the arc?

Problem 6

A circle with a shaded sector is shown.

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13) What is the area of the shaded sector? Write your answer in terms of π\pi.

14) What is the length of the arc that outlines this sector? Write your answer in terms of π\pi.

Problem 7

The towns of Washington, Franklin, and Springfield are connected by straight roads. The towns wish to build an airport to be shared by all of them.

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15) Where should they build the airport if they want it to be the same distance from each town’s center?

a)

At the midpoint of side WSWS.

b)

At the midpoint of side FSFS.

c)

At the midpoint of side WFWF.

d)

At the incenter of triangle WFSWFS.

e)

At the circumcenter of triangle WFSWFS.

16) Describe how to find the precise location.

17) Where should they build the airport if they want it to be the same distance from each of the roads connecting the towns?

a)

At the midpoint of side WSWS.

b)

At the midpoint of side FSFS.

c)

At the midpoint of side WFWF.

d)

At the incenter of triangle WFSWFS.

e)

At the circumcenter of triangle WFSWFS.

f)


18) Describe how to find the precise location.

Problem 8

19) Chords ACAC and DBDB intersect at point EE. Select all\textbf{all} pairs of angles that must be congruent. Write each corresponding letter in the answer box and separate letters with commas.

a) angle ADBADB and angle ACBACB \quad\quad b) angle ADBADB and angle CADCAD \quad\quad c) angle DEADEA and angle CEBCEB

d) angle CADCAD and angle CBDCBD \quad\quad e) angle BCABCA and angle CBACBA

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