Question

ky found the following ratios
\\( \frac { p q } { e f } = \frac { 9 } { 18 } \\)
\\( \frac { q r } { f g } = \frac { 15 } { 30 } \\)
\\( \frac { r p } { g e } = \frac { 19 } { 38 } \\)
is \\( \triangle p q r \sim \triangle e f g \\) ?
no, the triangles are not similar
yes, the triangles are similar by the sss similarity
theorem
there is not enough information to determine
whether the triangles are similar

Explanation:

Step1: Simplify the ratios

Simplify each ratio:

• $\frac{PQ}{EF}=\frac{9}{18}=\frac{1}{2}$
• $\frac{QR}{FG}=\frac{15}{30}=\frac{1}{2}$
• $\frac{RP}{GE}=\frac{19}{38}=\frac{1}{2}$

Step2: Check the correspondence

In $\triangle PQR$ and $\triangle EFG$, for SSS similarity, the sides must be in proportion and correspond. But in $\triangle PQR$, the sides are $PQ = 9$, $QR=15$, $RP = 19$ and in $\triangle EFG$, the sides are $EF = 18$, $FG=30$, $GE = 38$. The correspondence of sides is not correct for similarity (e.g., if $\triangle PQR\sim\triangle EFG$, we would need $\frac{PQ}{EF}=\frac{QR}{FG}=\frac{RP}{GE}$ with correct vertex - to - vertex correspondence. Here, if we assume $\triangle PQR\sim\triangle EFG$, the side - to - side match is wrong as per triangle similarity rules). So the triangles are not similar.

Answer:

A. No, the triangles are not similar