Question

find rs.
rs =
submit

Explanation:

Step1: Use the mid - segment theorem

The mid - segment theorem states that the mid - segment of a triangle is parallel to the third side and half its length. In triangle \(RUT\), \(SV\) is the mid - segment. So, \(SV=\frac{1}{2}RU\). But we can also use the similarity of triangles. Another approach: since \(V\) is the mid - point of \(RU\) (\(RV = 28\), \(RU=RV + VU\), assume \(RV = VU\)) and \(S\) is a point such that \(VS\parallel UT\). Also, we can use the property of similar triangles. Let's use the basic proportionality theorem (Thales' theorem). But more straightforwardly, consider the fact that if we assume two similar triangles (by AA similarity, as corresponding angles are equal due to parallel lines). Let's use the ratio of sides.
We know that in a triangle, if a line is parallel to one side and intersects the other two sides, it divides them proportionally. Here, assume \(\triangle RVS\sim\triangle RUT\) (by AA similarity, \(\angle R\) is common and \(\angle RVS=\angle RUT\) if \(VS\parallel UT\)). The ratio of sides: \(\frac{RS}{RT}=\frac{RV}{RU}\). But another way, since \(VS\) is a mid - segment (assuming \(V\) is the mid - point of \(RU\) and \(S\) is related to \(T\) in a proportionate way). Wait, actually, we can use the property that \(RS\) and \(ST\) are related. Wait, no, let's use the fact that \(VS\) is parallel to \(UT\) and \(V\) is the mid - point of \(RU\). Then \(S\) is the mid - point of \(RT\) (by the converse of the mid - segment theorem). So \(RS = ST\). Given \(ST = 48\).

Answer:

\(48\)