Question

verify that \\( \triangle p q r \sim \triangle s t u \\). determine the scale factor.
\\( \triangle p q r: p q=49, q r=84,\angle q=37^{\circ} \\)
\\( \triangle s t u: s t=21, t u=36,\angle t=37^{\circ} \\)
yes, with a scale factor of (7/3)
no, the scale is not proportional
yes, because the angles are equal
no, because the angles are equal

Explanation:

Step1: Check the ratio of corresponding sides

For \(\triangle PQR\) and \(\triangle STU\), we have \(PQ = 49\), \(QR=84\), \(ST = 21\), \(TU = 36\).
The ratio of \(PQ\) to \(ST\) is \(\frac{PQ}{ST}=\frac{49}{21}=\frac{7}{3}\).
The ratio of \(QR\) to \(TU\) is \(\frac{QR}{TU}=\frac{84}{36}=\frac{7}{3}\).

Step2: Check the included angles

We are given that \(\angle Q = 37^{\circ}\) and \(\angle T=37^{\circ}\), so \(\angle Q=\angle T\).

Since the ratios of the corresponding sides are equal (\(\frac{PQ}{ST}=\frac{QR}{TU}=\frac{7}{3}\)) and the included angles (\(\angle Q\) and \(\angle T\)) are equal, by the Side - Angle - Side (SAS) similarity criterion, \(\triangle PQR\sim\triangle STU\). The scale factor is the ratio of the corresponding sides, which is \(\frac{7}{3}\).

Answer:

Yes, with a scale factor of \(\frac{7}{3}\) (the first option).