Question

Consider the diagram shown and answer the following questions; the radius of this circle is 6 inches:

Answer 1

Refer to the figure shown below.

Part 1
Triangles  VPU and SPU are congruent because of SSA. Therefore,
∠TUV = ∠TUS = 39°.
Because angles in quadrilateral PVUS sum to 360°, therefore
The Central ∠VPS = 360 -(90+39+39+90) = 102°.
The measure of arc VTS = ∠VPS = 102°.

Part 2 
We know that line segments  PO = PS = PT = 6 in (radius).
Because TO is a diameter, ∠TMO = 90°.
From the indicated right triangles ΔUPV, ΔUPS, ΔMOT, we can calculate the lengths of PU, US, UV, MT and MO.

Part 3
Angles in a triangle sum to 90°. Because ∠MOP = 49°, therefore
∠MTO = 90 - 49 = 41°.
∠VPU = ∠SPU = 90 - 39 = 51°.