Question

determine if triangle pqr and triangle stu are or are not similar, and, if they are, state how you know. (note that figures are not necessarily drawn to scale.)

Explanation:

Step1: Calculate the ratios of corresponding sides

For triangle \(PQR\) with sides \(PQ = 19\), \(QR=19\), \(PR = 25\) and triangle \(STU\) with sides \(ST = 76\), \(TU = 76\), \(SU=100\)
The ratio of \(PQ\) to \(ST\) is \(\frac{PQ}{ST}=\frac{19}{76}=\frac{1}{4}\)
The ratio of \(QR\) to \(TU\) is \(\frac{QR}{TU}=\frac{19}{76}=\frac{1}{4}\)
The ratio of \(PR\) to \(SU\) is \(\frac{PR}{SU}=\frac{25}{100}=\frac{1}{4}\)

Step2: Check the similarity criterion

Since \(\frac{PQ}{ST}=\frac{QR}{TU}=\frac{PR}{SU}=\frac{1}{4}\), by the Side - Side - Side (SSS) similarity criterion, two triangles are similar if the ratios of their corresponding sides are equal.

Answer:

The triangles \(PQR\) and \(STU\) are similar.