Final answer:
By applying the Midsegment Theorem to the trapezoid and setting the segment connecting the midpoints equal to half the sum of the lengths of the two bases, we derived an equation and solved it to find that the value of t is 61.
Step-by-step explanation:
If x and U are the midpoints of the legs of a trapezoid, then by the Midsegment Theorem, the length of the segment connecting these midpoints is equal to half the sum of the lengths of the two bases of the trapezoid. In this case, the two bases are VW and YZ. We can set up the equation:
UX = \( \frac{1}{2}(VW + YZ) \).
Given VW = t-36, UX = -t+94, and YZ = 2t-81, we can substitute these values into the equation:
-t + 94 = \( \frac{1}{2}((t-36) + (2t-81)) \).
Multiplying both sides by 2 to clear the fraction and then simplifying, we get:
-2t + 188 = t - 36 + 2t - 81,
Now, summarizing like terms, we have:
-2t + 188 = 3t - 117.
Adding 2t to both sides gives:
188 = 5t - 117.
Finally, adding 117 to both sides and then dividing by 5, we get:
t = \( \frac{305}{5} \) = 61.
Thus, the value of t is 61.