Question

ky found the following ratios.
\\(\frac{pq}{ef} = \frac{9}{18}\\)
\\(\frac{qr}{fg} = \frac{15}{30}\\)
\\(\frac{rp}{ge} = \frac{19}{38}\\)
is \\(\triangle pqr \sim \triangle efg\\)?
\\(\circ\\) no, the triangles are not similar.
\\(\circ\\) yes, the triangles are similar by the sss similarity theorem.
\\(\circ\\) there is not enough information to determine whether the triangles are similar.

Explanation:

Step1: Simplify each ratio

Simplify \(\frac{PQ}{EF}=\frac{9}{18}\), divide numerator and denominator by 9: \(\frac{9\div9}{18\div9}=\frac{1}{2}\).
Simplify \(\frac{QR}{FG}=\frac{15}{30}\), divide numerator and denominator by 15: \(\frac{15\div15}{30\div15}=\frac{1}{2}\).
Simplify \(\frac{RP}{GE}=\frac{19}{38}\), divide numerator and denominator by 19: \(\frac{19\div19}{38\div19}=\frac{1}{2}\).

Step2: Apply SSS similarity theorem

The SSS (Side - Side - Side) similarity theorem states that if the corresponding sides of two triangles are in proportion, then the triangles are similar.
For \(\triangle PQR\) and \(\triangle EFG\), we have \(\frac{PQ}{EF}=\frac{QR}{FG}=\frac{RP}{GE}=\frac{1}{2}\), which means the corresponding sides are proportional. So by the SSS similarity theorem, \(\triangle PQR\sim\triangle EFG\).

Answer:

Yes, the triangles are similar by the SSS similarity theorem.