Question

given that \\(\frac{df}{pr} = \frac{fe}{rq} = \frac{3}{2}\\), what additional information is needed to prove \\(\triangle def \sim \triangle pqr\\) using the sss similarity theorem?\
\\(\bigcirc\\) \\(de \cong pq\\)\
\\(\bigcirc\\) \\(\angle d \cong \angle p\\)\
\\(\bigcirc\\) \\(\frac{de}{ef} = \frac{3}{2}\\)\
\\(\bigcirc\\) \\(\frac{de}{pq} = \frac{3}{2}\\)

Explanation:

Step1: Recall SSS Similarity Theorem

The SSS (Side - Side - Side) similarity theorem states that if the corresponding sides of two triangles are in proportion, then the two triangles are similar. For two triangles \(\triangle DEF\) and \(\triangle PQR\) to be similar by SSS similarity, the ratios of all three pairs of corresponding sides must be equal.

We are given that \(\frac{DF}{PR}=\frac{FE}{RQ}=\frac{3}{2}\). We already have the ratios of two pairs of corresponding sides (\(DF\) and \(PR\); \(FE\) and \(RQ\)). We need the ratio of the third pair of corresponding sides (\(DE\) and \(PQ\)) to also be \(\frac{3}{2}\) so that \(\frac{DF}{PR}=\frac{FE}{RQ}=\frac{DE}{PQ}=\frac{3}{2}\).

Step2: Analyze Each Option

• Option 1: \(DE\cong PQ\) means \(DE = PQ\), so the ratio \(\frac{DE}{PQ}=1\), which is not equal to \(\frac{3}{2}\). So this option is incorrect.
• Option 2: \(\angle D\cong\angle P\) is related to the SAS (Side - Angle - Side) similarity or congruence, not SSS similarity. So this option is incorrect.
• Option 3: \(\frac{DE}{EF}=\frac{3}{2}\) gives the ratio of two sides within \(\triangle DEF\), not the ratio of the corresponding sides of \(\triangle DEF\) and \(\triangle PQR\). So this option is incorrect.
• Option 4: \(\frac{DE}{PQ}=\frac{3}{2}\) gives the ratio of the third pair of corresponding sides. When combined with the given \(\frac{DF}{PR}=\frac{FE}{RQ}=\frac{3}{2}\), we have \(\frac{DF}{PR}=\frac{FE}{RQ}=\frac{DE}{PQ}=\frac{3}{2}\), which satisfies the SSS similarity theorem.

Answer:

\(\boldsymbol{\frac{DE}{PQ}=\frac{3}{2}}\) (the fourth option: \(\frac{DE}{PQ}=\frac{3}{2}\))