# Lesson 9

Part to Whole

### Problem 1

Jada cuts out a rectangular piece of paper that measures 5 inches by 4 inches. Han cuts out a paper sector of a circle with radius 5 inches, and calculates the arc length to be $$2\pi$$ inches. Whose paper is larger? Show your reasoning.

### Problem 2

A circle has radius 10 centimeters. Suppose an arc on the circle has length $$8\pi$$ centimeters. What is the measure of the central angle whose radii define the arc?

### Problem 3

A circle has radius 6 units. For each arc length, find the area of a sector of this circle which defines that arc length.

1. $$4\pi$$ units
2. $$5\pi$$ units
3. 10 units
4. $$\ell$$ units

### Problem 4

Select all the sectors which have an area of $$3\pi$$ square units.

A:

a sector with a radius of 6 units and a central angle of 30 degrees

B:

a sector with a radius of 6 units and a central angle of 45 degrees

C:

a sector with a radius of 3 units and a central angle of 60 degrees

D:

a sector with a radius of 3 units and a central angle of 120 degrees

E:

a sector with a radius of 3 units and a central angle of 180 degrees

### Solution

(From Unit 7, Lesson 8.)

### Problem 5

A circle has radius 4 units and a central angle measuring 45 degrees. What is the length of the arc defined by the central angle?

### Solution

(From Unit 7, Lesson 8.)

### Problem 6

Clare and Diego are discussing inscribing circles in quadrilaterals.

Diego thinks that you can inscribe a circle in any quadrilateral since you can inscribe a circle in any triangle. Clare thinks it is not always possible because she does not think the angle bisectors are guaranteed to intersect at a single point. She claims she can draw a quadrilateral for which an inscribed circle can’t be drawn.

Do you agree with either of them? Explain or show your reasoning.

### Solution

(From Unit 7, Lesson 7.)

### Problem 7

Triangle $$ABC$$ is shown together with the angle bisectors of each of its angles. Draw a point $$D$$ that is equidistant from sides $$AC$$ and $$AB$$, but which is closest to side $$BC$$.

### Solution

(From Unit 7, Lesson 6.)

### Problem 8

Priya and Mai are trying to prove that if 2 chords are congruent, they are equidistant from the center of the circle. Priya draws this picture.

Mai adds the perpendicular segment from the center of the circle to each chord.