Final answer:
None of the given options are correct for determining the perimeter of triangle RST. Assuming TS = 64, VW = 10, and UV = 32 (inferred from midsegment properties), the perimeter of triangle RST would be 116.
Step-by-step explanation:
In the context of a triangle with midsegments, the midsegments connect the midpoints of two sides of the triangle and are parallel to the third side. The length of a midsegment is half the length of the third side it is parallel to. In this problem, we are given the lengths of two midsegments (UW and VW) of triangle RST and one side of the triangle (TS).
Since UW is a midsegment parallel to TS, UW is half the length of TS, which makes TS twice the length of UW: TS = 2 * UW = 2 * 28 = 56. However, the question states that TS = 64, so this indicates a typo in the question. Assuming TS indeed equals to 64, then UV is the midsegment parallel to TS. Using the property of midsegments, we can say UV = TS / 2 = 64 / 2 = 32.
Similarly, VW is a midsegment parallel to another side of the triangle (let's call it RS). Thus, RS has to be double the length of VW: RS = 2 * VW = 2 * 10 = 20.
Finally, since the perimeter of a triangle is the sum of all its sides, we can add up the lengths of TS, RS, and RT (which is equal to UV since UV is the midsegment parallel to RT): Perimeter of ∆RST = TS + RS + RT = 64 + 20 + 32 = 116. Therefore, none of the options provided are correct choices for the perimeter of ∆RST.