Question

If TR = 11 ft, find the length of PS. Round to the nearest hundredth.PTR16°Sarc PS =ft

Answer 1

Given:

TR = 11 Ft

∠RTS = 16°

Let's determine the length of Arc PS,

Step 1: Let's first determine the angle of Arc PS.

∠RTS = 16°

∠PTQ = 16° ; Vertical angle pair of ∠RTS

Since ∠QTR = ∠PTS under the rule of vertical angles, we can now determine the measure of ∠PTS or Arc PS.

We get,

$\angle RTS\text{ + }\angle PTQ\text{ + }\angle QTR\text{ + }\angle PTS=360$$16\text{ + }16\text{ + }\angle PTS\text{ + }\angle PTS=360$$32\text{ + }2\angle PTS=360^{}$$\angle PTS=\frac{360^{}\text{ - 32}}{2}$$\angle PTS=\frac{328}{2}$$\angle PTS=164^{\circ}$

Step 2: Let's determine the perimeter of the circle.

$\text{ Perimeter = }2\pi r$$\text{ = 2}\pi(11)$$\text{ Perimeter = 22}\pi$

Step 3: Let's determine the length of Arc PS.

$\text{ Arc Length = (}\frac{\theta}{360})(\text{Perimeter of the Circle)}$$\text{ = (}\frac{164}{360})(22)(3.14)$$\text{ Arc Length = }31.469777\ldots\text{ }\approx\text{ 31.47 ft.}$

Therefore, the length of Arc PS is 31.47 ft.