Question

consider one triangle whose sides measure \frac{3}{2} units, \frac{5}{2} units, and 2 units. consider another triangle whose sides measure 2 units, \frac{5}{2} units, and \frac{3}{2} units. are these triangles congruent, similar, or both? explain your answer.

Explanation:

Step1: Recall the definitions

Congruent triangles have all corresponding sides equal. Similar triangles have all corresponding sides in proportion.

Step2: Check for congruence

The first triangle has sides \(\frac{3}{2}\), \(\frac{5}{2}\), \(2\). The second triangle has sides \(2\), \(\frac{5}{2}\), \(\frac{3}{2}\). By the SSS (Side - Side - Side) congruence criterion, since all three pairs of corresponding sides are equal (\(\frac{3}{2}=\frac{3}{2}\), \(\frac{5}{2}=\frac{5}{2}\), \(2 = 2\)), the triangles are congruent.

Step3: Check for similarity (as a consequence)

Since congruent triangles have a side - ratio of \(1\) (because \(\frac{\frac{3}{2}}{\frac{3}{2}}=\frac{\frac{5}{2}}{\frac{5}{2}}=\frac{2}{2}=1\)), they also satisfy the condition for similarity (where the ratio of corresponding sides is constant).

Answer:

The triangles are both congruent and similar. Because by the SSS congruence criterion (all corresponding sides are equal: \(\frac{3}{2}=\frac{3}{2}\), \(\frac{5}{2}=\frac{5}{2}\), \(2 = 2\)), they are congruent. And since congruent triangles have a ratio of \(1\) for all corresponding side - pairs, they also meet the similarity condition (ratio of corresponding sides is constant).