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Section 1
Circumference and Arc Length
COMPLETE THE SENTENCE The circumference of a circle with diameter d is $C=$ ____
WRITING Describe the difference between an arc measure and an arc length.
In Exercises $3-10,$ Find the indicated measure.(See Examples 1 and 2 .)circumference of a circle with a radius of 6 inches
In Exercises $3-10,$ Find the indicated measure.(See Examples 1 and 2 .)diameter of a circle with a circumference of 63 feet
In Exercises $3-10,$ Find the indicated measure.(See Examples 1 and 2 .)radius of a circle with a circumference of 28$\pi$
In Exercises $3-10,$ Find the indicated measure.(See Examples 1 and 2 .)exact circumference of a circle with a diameter of 5 inches
In Exercises $3-10,$ Find the indicated measure.(See Examples 1 and 2 .)arc length of $\overline{A B}$
In Exercises $3-10,$ Find the indicated measure.(See Examples 1 and 2 .)$\mathrm{mDE}$
In Exercises $3-10,$ Find the indicated measure.(See Examples 1 and 2 .)circumference of $\odot \mathrm{C}$
In Exercises $3-10,$ Find the indicated measure.(See Examples 1 and 2 .)radius og $\odot R$
ERROR ANALYSIS Describe and correct the error in finding the circumference of $\odot \mathrm{C}$ .
ERROR ANALYSIS Describe and correct the error in finding the length of \mathrm{GH}$ .
PROBLEM SOLVINGA measuring wheel is used to calculate the length of a path. The diameter of the wheel is 8 inches. The wheel makes 87 complete revolutions along the length of the path. To the nearest foot, how long is the path? (See Example 3.)
PROBLEM SOLVING You ride your bicycle 40 meters.How many complete revolutions does the front wheel make?
In Exercises $15-18,$ Ind the perimeter of the shaded region. (See Example $4 . )$
In Exercises $19-22,$ convert the angle measure.(See Example. 5.)Convert $70^{\circ}$ to radians.
In Exercises $19-22,$ convert the angle measure.(See Example. 5.)Convert $300^{\circ}$ to radians.
In Exercises $19-22,$ convert the angle measure.(See Example. 5.)Convert $\frac{11 \pi}{12}$ radians to degrees.
In Exercises $19-22,$ convert the angle measure.(See Example. 5.)Convert $\frac{\pi}{8}$ radian to degrees.
PROBLEM SOLVING The London Eye is a Ferris wheel in London, England, that travels at a speed of 0.26 meter per second. How many minutes does it take the London Eye to complete one full revolution?
PROBLEM SOLVING You are planning to plant a circular garden adjacent to one of the comers of abuilding, as shown. You can use up to 38 feet of fence to make a border around the garden. What radius(in feet) can the garden have? Choose all that apply. Explain your reasoning.
In Exercises 25 and $26,$ Find the circumference of the circle with the given equation. Write the circumference in terms of $\pi .$$x^{2}-y^{2}=16$
In Exercises 25 and $26,$ Find the circumference of the circle with the given equation. Write the circumference in terms of $\pi .$$(x=2)^{2}=(y-3)^{2}=9$
USING STRUCTURE A semicircle has endpoints $(-2,-5)$ and $(2,8) .$ Find the arc length of the semicircle.
REASONING EF is an arc on a circle with radius $\mathrm{r}$ . Let $\mathrm{x}^{\circ}$ be the measure of EF Describe the effect on the length of EF if you (a) double the radius of the circle, and (b) double the measure of EF.
MAKING AN ARGUMENT Your friend claims that it is possible for two arcs with the same measure to have different arc lengths. Is your friend correct? Explain your reasoning.
PROBLEM SOLVING Over 2000 years ago, the Greek scholar Eratosthenes estimated Earth's circumference by assuming that the Sun's rays were parallel. He chose a day when the Sun shone straight down into a well in the city of Syene. At noon, he measured the angle the Sun's rays made with a vertical stick in the city of Alexandria. Eratosthenes assumed that the distance from Syene to Alexandria was equal to about 575 miles. Explain how Eratosthenes was able to use this information to estimate Earth's circumference. Then estimate Earth's circumference.
ANALYZMG RELATIONSHIPS $\odot C$ , the ratio of the length of $\mathrm{PQ}$ to the length of RS is 2 to $1 .$ What is the ratio of $m \angle \mathrm{PCQ}$ to $\mathrm{mL} \mathrm{RCS}$ ?$\begin{array}{lll}{{A}} & {4 \text { to } 1} & {} & {{B}} & {2 \text { to } 1} \\ {C} & {1 \text { to } 4} & {} & {{D}} & {1 \text { to } 2}\end{array}$
ANALYZING RELATIONSHIPS A $45^{\circ}$ arc in $\odot \mathrm{C}$ and a length of $\mathrm{PQ}$ to the length of RS is 2 to $1 .$ What is the ratio of $m \angle \mathrm{PCQ}$ to $\mathrm{mL} \mathrm{RCS}$ ?
PROBLEM SOLVING HOW many revolutions does the smaller gear complete during a single revolution ofthe larger gear?
USING STRUCTURE ind the circumference of each circle.a. a circle circumscribed about a right triangle whose legs are 12 inches and 16 inches longb. a circle circumscribed about a square with a side length of 6 centimetersc. a circle inscribed in an equilateral triangle with a side length of 9 inches
REWRITING A FORMULA Write a formula in terms of the measure $\theta$ (theta) of the central angle (in radians) that can be used to Ind the length of an arc of a circle. Then use this formula to Ind the length of an arc of a circle with a radius of 4 inches and a central angle of $\frac{3 \pi}{4}$ radians.
HOW DO YOU SEE IT?Compare the circumference of $\odot \mathrm{P}$ to the length of DE. Explain your reasoning.
MAKING AN ARGUMENT In the diagram, the measure of the red shaded angle is $30^{\circ}$ . The arc length a is 2 . Your classmate claims that it is possible to and the circumference of the blue circle without Inding the radius of either circle. Is your classmate correct? Explain your reasoning.
MODELING WITH MATHEMATICS What is the measure (in radians) of the angle formed by the hands of a clock at each time? Explain your reasoning.$\text{a. 1.30} \quad \text{b. $3 : 15$}$
MATHEMATICAL CONNECTION The sum circumferences of circles $A B,$ and $C$ is 63$\pi$ . Find AC.
THOUGHT PROVOKING Is $\pi$ a rational number? Compare the rational number $\frac{355}{113}$ to $\pi$ . Find a different rational number that is even closer to $\pi$ .
PROOF The circles in the diagram are concentric and $\overline{\mathrm{FG}} \cong \overline{\mathrm{GH}}$ . Prove that $\mathrm{HK}$ and $\mathrm{NG}$ have the same length.
REPEATED REASONING $\overline{A B}$ is divided into four congruent segments, and semicircles with radius r are drawn.a. What is the sum of the four arc lengths?b. What would the sum of the arc lengths be if AB was divided into 8 congruent segments? 16 -congruent segments h congruent segments? Explain your reasoning.
Find the area of the polygon with the given vertices.$\mathrm{X}(2,4), \mathrm{Y}(8,-1), \mathrm{Z}(2,-1)$
Find the area of the polygon with the given vertices.$\mathrm{L}(-3,1), \mathrm{M}(4,1), \mathrm{N}(4,-5), \mathrm{P}(-3,-5)$