Question

in circle r, rs = 14 and the length of \\(\overarc{st}\\) = 7\\(\pi\\). find m\\(\angle srt\\).

Explanation:

Step1: Recall arc length formula

The formula for the length of an arc \( s \) is \( s=\frac{\theta}{360^\circ}\times2\pi r \), where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle. Here, \( RS \) is the radius, so \( r = 14 \), and the arc length \( s=\text{length of } \overarc{ST}=7\pi \).

Step2: Substitute values into the formula

Substitute \( s = 7\pi \), \( r = 14 \) into \( s=\frac{\theta}{360^\circ}\times2\pi r \):
\[
7\pi=\frac{\theta}{360^\circ}\times2\pi\times14
\]

Step3: Solve for \( \theta \)

First, simplify the right - hand side: \( \frac{\theta}{360^\circ}\times28\pi=\frac{28\pi\theta}{360^\circ} \)
The equation becomes \( 7\pi=\frac{28\pi\theta}{360^\circ} \)
Divide both sides by \( \pi \): \( 7=\frac{28\theta}{360^\circ} \)
Multiply both sides by \( 360^\circ \): \( 7\times360^\circ = 28\theta \)
\( 2520^\circ=28\theta \)
Divide both sides by 28: \( \theta=\frac{2520^\circ}{28}=90^\circ \)

Answer:

\( 90^\circ \)