Question

find ru.
ru =

Explanation:

Step1: Use the Mid - segment Theorem

The Mid - segment Theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. Here, \(VT\) is the mid - segment of \(\triangle RSU\) (since \(V\) is the midpoint of \(RU\) and \(T\) is the midpoint of \(SU\)).

Step2: Calculate \(SU\)

We know that \(SU=16 + 32=48\).

Step3: Apply the Mid - segment formula

Since \(VT=\frac{1}{2}RS\) (not relevant here) and \(RU\) can be found using the fact that in the trapezoid - like figure (or using the property of mid - segments in a different geometric interpretation, if we consider the lines and mid - points), we can also use the fact that if we assume the lines \(RU\) and \(TS\) are related through the mid - point \(V\) and \(T\). Another approach:
We know that \(RU\) and \(TS\) are related. Since \(V\) is the mid - point of \(RU\) (by the mark on the line segment) and \(T\) is the mid - point of \(SU\) (by the mark on the line segment).
We can use the formula for the length of \(RU\) in terms of the parallel segments. If we consider the fact that \(RU\) and \(TS\) are related in a way that \(RU = 2\times25\) (because of the mid - point and the parallel - like property in the geometric figure, assume \(RU\) is composed of two segments of length \(25\) each)

Answer:

\(RU = 50\)