Question

in circle c with m∠bcd = 60° and bc = 17 units, find the length of \\(\overarc{bd}\\). round to the nearest hundredth.

Explanation:

Step1: Recall arc length formula

The formula for the length of an arc \( \widehat{BD} \) in a circle is \( s = r\theta \), where \( r \) is the radius and \( \theta \) is the central angle in radians. First, we need to convert the central angle from degrees to radians. We know that \( 60^\circ=\frac{\pi}{3} \) radians (since to convert degrees to radians, we use \( \theta_{rad}=\theta_{deg}\times\frac{\pi}{180} \), so \( 60\times\frac{\pi}{180}=\frac{\pi}{3} \)). Also, the radius \( r = BC = 17 \) units (since \( BC \) is a radius of the circle).

Step2: Calculate arc length

Substitute \( r = 17 \) and \( \theta=\frac{\pi}{3} \) into the arc length formula \( s = r\theta \). So \( s=17\times\frac{\pi}{3} \). Calculate this value: \( 17\times\frac{\pi}{3}\approx17\times1.0472\approx17.8024 \). Rounding to the nearest hundredth, we get approximately \( 17.80 \).

Answer:

\( 17.80 \)