Question

1. the circumference of the circle centered at point s is 136 millimeters. the length of arc egf is 119 millimeters. find the measure of ∠esf

1. a circle centered at point u is shown. if the length of arc wv is 33 inches, find the circumference of the circle.

1. a circle centered at point o is shown. the length of arc ap is \\(\frac{3}{8}\\) of the circumference of the circle. what is the value of x?

Explanation:

Problem 3

Step1: Find arc length of \( \overset{\frown}{EF} \)

Circumference \( C = 136 \) mm, arc \( \overset{\frown}{EGF} = 119 \) mm. So arc \( \overset{\frown}{EF}=136 - 119 = 17 \) mm.

Step2: Relate arc length to central angle

The ratio of arc \( \overset{\frown}{EF} \) to circumference is \( \frac{17}{136}=\frac{1}{8} \). A full circle is \( 360^\circ \), so central angle \( \angle ESF = 360^\circ\times\frac{1}{8}=45^\circ \).

Step1: Set up arc length formula

Arc length \( L=\frac{\theta}{360^\circ}\times C \), where \( \theta = 108^\circ \), \( L = 33 \) inches.

Step2: Solve for circumference \( C \)

\( 33=\frac{108^\circ}{360^\circ}\times C \). Simplify \( \frac{108}{360}=\frac{3}{10} \), so \( C = 33\div\frac{3}{10}=33\times\frac{10}{3}=110 \) inches.

Step1: Relate arc length ratio to central angle

Arc \( \overset{\frown}{AP} \) is \( \frac{3}{8} \) of circumference, so central angle \( x^\circ = 360^\circ\times\frac{3}{8} \).

Step2: Calculate \( x \)

\( 360\times\frac{3}{8}=135 \), so \( x = 135 \).

Answer:

\( 45^\circ \)

Problem 4