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 \(\frac{PQ}{EF}=\frac{9}{18}=\frac{1}{2}\), \(\frac{QR}{FG}=\frac{15}{30}=\frac{1}{2}\), \(\frac{RP}{GE}=\frac{19}{38}=\frac{1}{2}\)

Step2: Apply SSS similarity theorem

Since \(\frac{PQ}{EF}=\frac{QR}{FG}=\frac{RP}{GE}=\frac{1}{2}\), by the SSS (Side - Side - Side) similarity theorem, if the ratios of the corresponding sides of two triangles are equal, then the triangles are similar.

Answer:

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