Question

find tv.
tv =

Explanation:

Step1: Use the Mid - segment Theorem

The Mid - segment Theorem states that if a segment (in this case \(RU\)) is a mid - segment of a triangle (\(\triangle STV\), where \(RU\parallel ST\) and \(R\) is the mid - point of \(SV\) and \(U\) is the mid - point of \(TV\)), then \(RU=\frac{1}{2}ST\) and \(\frac{VR}{VS}=\frac{VU}{VT}\).

We know that \(VS = VR+RS=20 + 30=50\), \(VR = 20\), and \(VU = 12\). Let \(TV=x\), then \(VT=VU + UT\) and since \(U\) is the mid - point (because of the mid - segment \(RU\)), \(UT = VU = 12\) (not the correct way, we use the ratio).

By the basic proportionality theorem (Thales' theorem) in \(\triangle STV\) with \(RU\parallel ST\), we have \(\frac{VR}{VS}=\frac{VU}{VT}\).

Substitute \(VR = 20\), \(VS=50\), and \(VU = 12\) into the proportion \(\frac{VR}{VS}=\frac{VU}{VT}\).

We get \(\frac{20}{50}=\frac{12}{TV}\).

Step2: Solve the proportion for \(TV\)

Cross - multiply the proportion \(\frac{20}{50}=\frac{12}{TV}\).

We have \(20\times TV=50\times12\).

\(20TV = 600\).

Divide both sides by 20: \(TV=\frac{600}{20}\).

Answer:

\(30\)