Question

find st.
st =

Explanation:

Step1: Use the mid - segment theorem

The mid - segment theorem states that a segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. Here, \(SU\) is parallel to \(RV\) (by the arrow marks indicating parallelism), and \(SU\) is the mid - segment of \(\triangle TRV\). So, \(TR = 2ST\) (since \(S\) is the mid - point of \(TR\)). Let \(ST=x\), then \(TR=x + 32\). Also, since \(SU\) is the mid - segment, \(TV=2TU\). Given \(TU = 9\), then \(TV=2\times9=18\) (this step is to confirm the mid - segment relation for the base, but the key relation is for the side).

Another way: Since \(SU\) is parallel to \(RV\), \(\triangle TSU\sim\triangle TRV\) (by the AA similarity criterion, as \(\angle T=\angle T\) (common angle) and \(\angle TSU=\angle TRV\) (corresponding angles for parallel lines \(SU\parallel RV\)). The ratio of similarity is \(\frac{ST}{TR}=\frac{TU}{TV}\). Since \(SU\) is the mid - segment (by the arrow marks on the sides indicating mid - points), \(TV = 2TU\) (mid - segment theorem for the base of the triangle). But we can also use the property of the mid - point of the side.

Since \(S\) is the mid - point of \(TR\) (by the mid - segment property, as \(SU\parallel RV\) and \(U\) is the mid - point of \(TV\)), we know that \(TR=2ST\). Let \(ST=x\), then \(x+32 = 2x\).

Step2: Solve the equation

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Answer:

\(32\)