Question

7 △rsp-△qtp

Explanation:

Step1: Use the property of similar triangles

Since \(\triangle RSP\sim\triangle QTP\), the ratios of corresponding sides are equal.
For the sides \(QR = 5\), \(RP=5\), \(TS = 4\), \(SP=x\), we have \(\frac{QR}{QP}=\frac{TS}{TP}\). But \(QP=QR + RP=5 + 5 = 10\), \(TP=TS+SP=4 + x\). Also, from the similarity \(\frac{RP}{QP}=\frac{SP}{TP}\), substituting \(RP = 5\), \(QP = 10\) gives \(\frac{5}{10}=\frac{x}{4 + x}\).
Cross - multiply: \(5(4 + x)=10x\).
Expand: \(20+5x = 10x\).
Subtract \(5x\) from both sides: \(20=10x - 5x\), so \(5x=20\), \(x = 4\).

For the sides \(QT = 12\), \(RS=y\), using the ratio \(\frac{RP}{QP}=\frac{RS}{QT}\). Since \(RP = 5\), \(QP = 10\), \(QT = 12\).
Substitute into the ratio \(\frac{5}{10}=\frac{y}{12}\).
Cross - multiply: \(10y=5\times12\).
\(10y = 60\), \(y = 6\).

Answer:

\(x = 4\), \(y = 6\)