Question

type the correct answer in each box.
△abc is similar to △pqr.
overline{ab} corresponds to overline{pq}, and overline{bc} corresponds to overline{qr}.
if the length of overline{ab} is 9 units, the length of overline{bc} is 12 units, the length of overline{ca} is 6 units, and the length of overline{pq} is 3 units, then the length of overline{qr} is
units and the length of overline{rp} is
units.

Explanation:

Step1: Find the scale factor

Since \(\triangle ABC\sim\triangle PQR\) and \(\overline{AB}\) corresponds to \(\overline{PQ}\), the scale factor \(k=\frac{PQ}{AB}\). Given \(AB = 9\) units and \(PQ=3\) units, then \(k=\frac{3}{9}=\frac{1}{3}\).

Step2: Find the length of \(\overline{QR}\)

Since \(\overline{BC}\) corresponds to \(\overline{QR}\) and \(BC = 12\) units. Using the property of similar triangles \(\frac{QR}{BC}=k\). Substitute \(k=\frac{1}{3}\) and \(BC = 12\) into the formula: \(QR=\frac{1}{3}\times BC\). So \(QR=\frac{1}{3}\times12 = 4\) units.

Step3: Find the length of \(\overline{RP}\)

Since \(\overline{CA}\) corresponds to \(\overline{RP}\) and \(CA = 6\) units. Using the property of similar triangles \(\frac{RP}{CA}=k\). Substitute \(k = \frac{1}{3}\) and \(CA=6\) into the formula: \(RP=\frac{1}{3}\times CA\). So \(RP=\frac{1}{3}\times6=2\) units.

Answer:

The length of \(\overline{QR}\) is \(4\) units and the length of \(\overline{RP}\) is \(2\) units.