Question

in the diagram, ( fh=\frac{1}{3}ac ) and ( 3fg = ab ). what additional information is necessary to prove that ( \triangle abc ) is similar to ( \triangle fgh ), using the sss similarity theorem?
( angle baccongangle gfh )
( angle abccongangle fgh )
( \frac{bc}{gh}=3 )
( \frac{bc}{gh}=\frac{1}{3} )

Explanation:

Step1: Recall SSS similarity theorem

The SSS (Side - Side - Side) similarity theorem states that if the ratios of the corresponding sides of two triangles are equal, then the two triangles are similar.
We are given \(FH=\frac{1}{3}AC\) (i.e., \(\frac{AC}{FH} = 3\)) and \(3FG=AB\) (i.e., \(\frac{AB}{FG}=3\)).
For \(\triangle ABC\) and \(\triangle FGH\) to be similar by SSS similarity theorem, we need \(\frac{BC}{GH}=3\) so that \(\frac{AB}{FG}=\frac{AC}{FH}=\frac{BC}{GH}=3\).

Answer:

\(\frac{BC}{GH}=3\)