Explanation :
To find the value of x, we need to use the information given about the trapezoid and its midpoints.
First, we know that U is the midpoint of TQ. This means that the segment TU is equal in length to the segment UQ.
Similarly, V is the midpoint of SR. This means that the segment SV is equal in length to the segment VR.
Given that TS = 5, and UV = 786, we can set up the following equations:
TU + UQ = TS
UV + VQ = SR
Replacing the known values into the equations:
TU + UQ = 5
786 + VQ = SR
Since U is the midpoint of TQ, TU = UQ. Similarly, since V is the midpoint of SR, SV = VR. Therefore, we can simplify the equations as follows:
TU + TU = 5
786 + SV = SR
2TU = 5
786 + SV = SR
Since we are given the lengths of TS and QR in terms of x, let's substitute those values in:
2(8x + 34) = 5 (Equation 1)
786 + (14x + 92) = SR (Equation 2)
Now we can solve these equations to find the value of x.
Simplifying Equation 1:
16x + 68 = 5
16x = 5 - 68
16x = -63
x = -63/16
Therefore, the value of x is -63/16.