Lesson 14
Putting It All Together
Problem 1
Jada is finding the area of a sector with an angle \(\frac{\pi}{4}\) radians and radius 8 units. She found the area of the whole circle, then found the fraction represented by the sector by dividing \(\frac{\pi}{4}\) by 360. She multiplied this fraction by the total circle area.
- Do you agree with Jada’s strategy? Explain your reasoning.
- Find the area of the sector.
Solution
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(From Unit 7, Lesson 13.)Problem 2
Which of these pizza slices gives the best value (the most pizza per dollar spent)?
a slice with a radius of 12 inches, central angle of 30\(^\circ\), and a cost of $3 per slice
a slice with a radius of 8 inches, central angle of 45\(^\circ\), and a cost of $2 per slice
a slice with a radius of 6 inches, central angle of \(\frac{\pi}{3}\) radians, and a cost of $2 per slice
a slice with a radius of 6 inches, central angle of \(\frac{\pi}{4}\) radians, and a cost of $1 per slice
Solution
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(From Unit 7, Lesson 13.)Problem 3
The circle in the image has been divided into congruent sectors. What is the measure of the central angle of the shaded region in radians?
Solution
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(From Unit 7, Lesson 12.)Problem 4
In the circle, sketch a central angle that measures \(\frac{5\pi}{3}\) radians.
Solution
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(From Unit 7, Lesson 12.)Problem 5
The image shows a circle with radius 5 units.
- Draw a 180 degree central angle (a diameter) in the circle. What is the length of the arc defined by this angle?
- Use the arc length and the radius to calculate the radian measure of 180 degrees.
- Calculate the radian measure of a 360 degree angle. Explain or show your reasoning.
Solution
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(From Unit 7, Lesson 11.)Problem 6
Complete the table. Each row represents a circle with a defined sector.
sector area | radius | central angle |
---|---|---|
\(5\pi\) cm^{2} | 5 cm | |
\(12\pi\) cm^{2} | 270 degrees | |
12 cm | 15 degrees |
Solution
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(From Unit 7, Lesson 9.)Problem 7
Several circles with central angles are described. Select all the circles for which the central angle defines arcs that have length \(6\pi\) units.
radius 6 units, central angle 180 degrees
radius 18 units, central angle 60 degrees
radius 12 units, central angle 90 degrees
radius 3 units, central angle 120 degrees
radius 4 units, central angle 270 degrees
Solution
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(From Unit 7, Lesson 8.)Problem 8
Triangle \(ABC\) is shown with its incenter at \(D\). The inscribed circle’s radius measures 2 units. The length of \(AB\) is 9 units. The length of \(BC\) is 10 units. The length of \(AC\) is 17 units.
- What is the area of triangle \(ABD\)?
- What is the area of triangle \(BCD\)?
Solution
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(From Unit 7, Lesson 7.)Problem 9
Noah makes 3 statements about the incenter of a triangle.
- To find the incenter of a triangle, you must construct all 3 angle bisectors.
- The incenter is always equidistant from the vertices of the triangle.
- The incenter is always equidistant from each side of the triangle.
For each statement, decide whether you agree with Noah. Explain your reasoning.
Solution
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(From Unit 7, Lesson 6.)Problem 10
Elena is writing notes about central angles in circles. Help her finish her notes by answering the questions.
- Where is the vertex of a central angle located in relation to the circle?
- What line segments related to circles are contained in the rays that form a central angle?
- How does the measure of a central angle relate to the measure of the arc it is associated with?
Solution
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(From Unit 7, Lesson 1.)