Secant and Tangent Angles in Circles, finding an arc length.

[Cj111]

View attachment 8161

I got 138.17, but that isn't correct. I don't know how to do it, since the only way I thought, gave me the wrong answer. Can anyone help?

 

[MarkFL]

I would call point $C$ the center of the circle. So, what must the measures of \(\displaystyle \angle CPQ\) and \(\displaystyle \angle CRQ\) be?

 

[MarkFL]

To follow up, since \(\displaystyle \overline{QP}\) and \(\displaystyle \overline{QR}\) are tangent to the circle, we must have \(\displaystyle \angle CPQ=\angle CRQ=90^{\circ}\). Since the sum of the interior angles of a quadrilateral is $360^{\circ}$, it follows then that:

\(\displaystyle 100x+81x-1=180\implies x=1\)

And hence, \(\displaystyle \overset{\frown}{PR}=100^{\circ}\)