Question

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

1) If angle SUT is 39 degrees, what does that tell us about angle TUV? What arc measure describes arc VTS? How can we make any assertions about these angle and arc measures?
2) Which line segments' length can be calculated based on knowing the radius length? Explain .
3) If angle MOP is 49 degrees, which other angle measures can we calculate? What arc measures can we calculate? Describe which lesson concepts allow us to make these calculations

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°.