Question

Find the indicated arc length. Round answer to nearest tenth.RPTR97°PS.XT125°PS = 28 feet

Answer 1

To find the length of arc, we can use the formula

`l=(\theta)/(360)*2\pi r`

The radius (r) is half of PS, hence

`\begin{gathered} r=(28)/(2) \\ r=14\text{ f}eet \end{gathered}`

To find the angle:

`\begin{gathered} \angle QXR=125-97\text{ (vertically opposite angles are equal)} \\ \angle QXR=28^(\circ) \end{gathered}`

The Sum of angles in a point will give us

`\begin{gathered} \angle QXR+\angle RXS+\angle SXT+\angle TXP+\angle QXP=360_{} \\ \text{But} \\ \angle QXP=\angle SXT\text{ (Vertically opposite angles are equal)} \\ \text{Therefore} \\ 28+97+\angle QXP+125+\angle QXP=360 \end{gathered}`

Collecting like terms, we have

`\begin{gathered} 2\angle QXP=360-97-28-125 \\ =110 \\ \angle QXP=(110)/(2)=55^(\circ) \end{gathered}`

Therefore, the angle of arc RPT can be given as

`\begin{gathered} \angle QXR+\angle QXP+\angle PXT \\ \theta=28+55+125=208^(\circ) \end{gathered}`

Therefore, we find the length of the arc given we have all the parameters needed.

Hence,

`\begin{gathered} l=(\theta)/(360)*2\pi r \\ =(208)/(360)*2*\pi*14 \\ l=50.8\text{ fe}et \end{gathered}`

Therefore, the length of the arc RPT is 50.8 feet to the nearest tenth.