Question

circle j has a radius of 10 units, and points k, m, and l are on the circle. angle kjl has a measure of 135°, as shown. what is the length, in units, of minor arc kl? round your answer to the nearest hundredth.

Explanation:

Step1: Recall arc length formula

The formula for the length of an arc of a circle is \( s = r\theta \), where \( r \) is the radius of the circle and \( \theta \) is the central angle in radians. First, we need to convert the given angle from degrees to radians.
To convert degrees to radians, we use the conversion factor \( \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \). Given \( \theta = 135^\circ \), so \( \theta_{radians} = 135 \times \frac{\pi}{180} = \frac{3\pi}{4} \) radians.

Step2: Substitute values into arc length formula

We know the radius \( r = 10 \) units and \( \theta = \frac{3\pi}{4} \) radians. Substituting these values into the arc length formula \( s = r\theta \), we get \( s = 10 \times \frac{3\pi}{4} \).
Simplifying this expression: \( s = \frac{30\pi}{4} = \frac{15\pi}{2} \approx \frac{15\times3.14159}{2} \approx 23.56 \) (rounded to the nearest hundredth).

Answer:

\( 23.56 \)