Question

create and solve two proportions to find the values of x and y.

Explanation:

Step1: Set up proportion for \(x\)

Since \(\triangle QPS\) and \(\triangle TRS\) are similar (by AA similarity, as \(\angle S\) is common and \(\angle QPS=\angle TRS\) if \(TR\parallel QP\)), we use the ratio of corresponding sides.
\(\frac{QR}{RS}=\frac{PT}{TS}\), \(QR = 9\), \(RS=3\), \(PT = 15\), \(TS=x\).
\(\frac{9}{3}=\frac{15}{x}\)

Step2: Solve for \(x\)

Cross - multiply: \(9x=3\times15\)
\(9x = 45\)
\(x=\frac{45}{9}=5\)

Step3: Set up proportion for \(y\)

Using the ratio of corresponding sides \(\frac{QP}{TR}=\frac{QS}{RS}\), \(QP = 12\), \(TR = y\), \(QS=9 + 3=12\), \(RS = 3\)
\(\frac{12}{y}=\frac{12}{3}\)

Step4: Solve for \(y\)

Cross - multiply: \(12\times3=12y\)
\(y=\frac{36}{12}=3\)

Answer:

\(x = 5\), \(y = 3\)