Question

determine if 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 \(DE = 18\), \(EF=24\), \(FD = 28\) and triangle \(GHI\) with sides \(GH = 45\), \(HI = 60\), \(IG=70\).
The ratio of \(DE\) to \(GH\) is \(\frac{DE}{GH}=\frac{18}{45}=\frac{2}{5}\).
The ratio of \(EF\) to \(HI\) is \(\frac{EF}{HI}=\frac{24}{60}=\frac{2}{5}\).
The ratio of \(FD\) to \(IG\) is \(\frac{FD}{IG}=\frac{28}{70}=\frac{2}{5}\).

Step2: Check the similarity criterion

Since the ratios of the corresponding sides of triangle \(DEF\) and triangle \(GHI\) are equal (\(\frac{DE}{GH}=\frac{EF}{HI}=\frac{FD}{IG}=\frac{2}{5}\)), by the Side - Side - Side (SSS) similarity criterion, the two triangles are similar.

Answer:

The triangles \(DEF\) and \(GHI\) are similar. We know this because the ratios of their corresponding sides are equal: \(\frac{DE}{GH}=\frac{18}{45}=\frac{2}{5}\), \(\frac{EF}{HI}=\frac{24}{60}=\frac{2}{5}\), \(\frac{FD}{IG}=\frac{28}{70}=\frac{2}{5}\), satisfying the SSS (Side - Side - Side) similarity criterion.