# Lesson 10

Angles, Arcs, and Radii

- Let’s analyze relationships between arc lengths, radii, and central angles.

### Problem 1

Here are 2 circles. The smaller circle has radius \(r\), circumference \(c\), and diameter \(d\). The larger circle has radius \(R\), circumference \(C\), and diameter \(D\). The larger circle is a dilation of the smaller circle by a factor of \(k\).

Using the circles, write 3 pairs of equivalent ratios. Find the value of each set of ratios you wrote.

### Problem 2

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.

- What is the scale factor of dilation?
- What is the length of the highlighted arc in circle A?
- What is the length of the highlighted arc in circle B?
- What is the ratio of the arc lengths?
- How does the ratio of arc length compare to the scale factor?

### 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 \(4\pi\) inches. Whose paper is larger? Explain or show your reasoning.

### Problem 5

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

### Problem 6

A circle with a shaded sector is shown.

- What is the area of the shaded sector?
- What is the length of the arc that outlines this sector?

### 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.

- Where should they build the airport if they want it to be the same distance from each town’s center? Describe how to find the precise location.
- Where should they build the airport if they want it to be the same distance from each of the roads connecting the towns? Describe how to find the precise location.

### Problem 8

Chords \(AC\) and \(DB\) intersect at point \(E\). Select **all** pairs of angles that must be congruent.

angle A D B and angle A C B

angle A D B and angle C A D

angle D E A and angle C E B

angle C A D and angle C B D

angle B C A and angle C B D