Question

determine whether triangle def and triangle ghi 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 \(\triangle DEF\) with sides \(12\), \(16\), \(16\) and \(\triangle GHI\) with sides \(48\), \(64\), \(64\).
The ratio of the first pair of corresponding sides: \(\frac{12}{48}=\frac{1}{4}\)
The ratio of the second pair of corresponding sides: \(\frac{16}{64}=\frac{1}{4}\)
The ratio of the third pair of corresponding sides: \(\frac{16}{64}=\frac{1}{4}\)

Step2: Check the similarity criterion

Since the ratios of all three pairs of corresponding sides are equal (\(\frac{1}{4}\)), by the Side - Side - Side (SSS) similarity criterion.

Answer:

\(\triangle DEF\) and \(\triangle GHI\) are similar. We know this because the ratios of their corresponding sides are equal (\(\frac{DE}{GH}=\frac{EF}{HI}=\frac{DF}{GI}=\frac{1}{4}\)), so by the SSS (Side - Side - Side) similarity criterion, the two triangles are similar.