Question

find the measure of \\(\overline{sr}\\)
(image of triangle with points p, q, r, s, t; lengths: pt=14, pq=7, pr=12, ts=20; angles at p and s are equal)
options:
a 12
b 24
c 36
d 10

Explanation:

Step1: Determine similar triangles

Since \(\angle PQR=\angle PSR\) (given) and \(\angle P\) is common, \(\triangle PQR\sim\triangle PSR\) by AA (Angle - Angle) similarity criterion.

Step2: Set up proportion

For similar triangles \(\triangle PQR\) and \(\triangle PSR\), we have the proportion \(\frac{PQ}{PT}=\frac{PR}{PS}\). Let \(SR = x\). First, \(PQ = 7\), \(PT=7 + 14=21\), \(PR = 12\), and \(PS=12 + x\). But using the side - side - side similarity ratio \(\frac{PQ}{PT}=\frac{PR}{PS}\) can also be written as \(\frac{PQ}{QR}=\frac{PT}{TS}\) (another form of similarity ratio). Or more simply, using \(\frac{PQ}{PT - PQ}=\frac{PR}{SR}\) (since \(\triangle PQR\sim\triangle PST\) in a different approach of similarity ratios).

We know that \(\frac{PQ}{QT}=\frac{PR}{SR}\) (by basic proportionality theorem for similar triangles). Substituting \(PQ = 7\), \(QT = 14\), \(PR = 12\)

$$ \frac{7}{14}=\frac{12}{SR} $$

Cross - multiply: \(7\times SR=14\times12\)

$$ SR=\frac{14\times12}{7} $$

$$ SR = 24 $$

Answer:

B. 24