In a circle with center O and radius 5, points A, B, and C lie on the circle. If the measu… — SAT

In a circle with center $O$ and radius $5$ , points $A$ , $B$ , and $C$ lie on the circle. If the measure of the inscribed angle $\angle ABC$ is $\frac{2\pi}{5}$ radians, what is the length of minor arc $AC$ ?

A
$\pi$
B
$2\pi$
$4\pi$ Answer
D
$8\pi$

Answer

4\pi

According to the Inscribed Angle Theorem, the measure of a central angle is twice the measure of an inscribed angle that subtends the same arc. Given that the inscribed angle

\angle ABC

has a measure of

\frac{2\pi}{5}

radians, the corresponding central angle

\angle AOC

has a measure of

2 \times \frac{2\pi}{5} = \frac{4\pi}{5}

radians. Using the formula for arc length,

s = r\theta

, where

r = 5

is the radius and

\theta = \frac{4\pi}{5}

is the central angle in radians, the length of minor arc

AC

5 \times \frac{4\pi}{5} = 4\pi

Step-by-Step Solution

Find the measure of the central angle

\angle AOC

that subtends the same minor arc

AC

as the inscribed angle

\angle ABC

The measure of central angle

\angle AOC

2 \times \frac{2\pi}{5} = \frac{4\pi}{5}

radians.

By the Inscribed Angle Theorem, the measure of a central angle subtending an arc is twice the measure of any inscribed angle subtending the same arc.

Calculate the length of minor arc

AC

using the formula

s = r\theta

The arc length is

s = 5 \times \frac{4\pi}{5} = 4\pi

The formula for the arc length of a circle is

s = r\theta

, where

r

is the radius and

\theta

is the central angle measure in radians.

Key Concept

Inscribed Angle Theorem and Arc Length in Radians

Question

Answer

Step-by-Step Solution

Key Concept

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