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 $\triangle pqr sim \triangle efg$?
$\bigcirc$ no, the triangles are not similar.
$\bigcirc$ yes, the triangles are similar by the sss similarity theorem.
$\bigcirc$ there is not enough information to determine whether the triangles are similar.
diagrams: $\triangle pqr$ with $pq=9$, $qr=15$, $pr=19$; $\triangle efg$ with $fg=30$, $ef=18$, $ge=38$

Explanation:

Step1: Simplify each ratio

Simplify \(\frac{PQ}{EF}=\frac{9}{18}\), we get \(\frac{9\div9}{18\div9}=\frac{1}{2}\).
Simplify \(\frac{QR}{FG}=\frac{15}{30}\), we get \(\frac{15\div15}{30\div15}=\frac{1}{2}\).
Simplify \(\frac{RP}{GE}=\frac{19}{38}\), we get \(\frac{19\div19}{38\div19}=\frac{1}{2}\).

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 two triangles are similar. Since \(\frac{PQ}{EF}=\frac{QR}{FG}=\frac{RP}{GE}=\frac{1}{2}\), the corresponding sides of \(\triangle PQR\) and \(\triangle EFG\) are in proportion.

Answer:

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