Question

qu || rt. find rs.
rs =

Explanation:

Step1: Use the basic proportionality theorem (Thales' theorem)

Since \( \overline{QU}\parallel\overline{RT}\), we have \(\triangle STR\sim\triangle SUQ\). Then \(\frac{ST}{SU}=\frac{SR}{SQ}\).
We know that \(SU = ST + TU=10 + 30=40\) and \(SQ=SR + RQ\). Let \(RS=x\), then \(SQ=x + 20\).

Step2: Set up the proportion equation

Substituting the values into \(\frac{ST}{SU}=\frac{SR}{SQ}\), we get \(\frac{10}{40}=\frac{x}{x + 20}\).
Cross - multiply: \(10(x + 20)=40x\).
Expand: \(10x+200 = 40x\).
Subtract \(10x\) from both sides: \(200=40x - 10x\).
Simplify: \(200 = 30x\).
Solve for \(x\): \(x=\frac{200}{30}=\frac{20}{3}\approx6.67\).

Answer:

\(\frac{20}{3}\)