Question

# Given /_ DEF and /_ EGF in the diagram below, determine if the triangles are similar. If so, write a similarity statement, and state the criterion used to support your claim.12210107071.jpg

Similarity

Given $$\displaystyle\triangle{D}{E}{F}{\quad\text{and}\quad}\triangle{E}{G}{F}$$ in the diagram below, determine if the triangles are similar. If so, write a similarity statement, and state the criterion used to support your claim.

2020-10-28
Step 1
Given:$$\displaystyle{D}{E}={5},{E}{F}={4},{F}{D}=\frac{{32}}{{5}},{F}{G}=\frac{{5}}{{2}},{E}{G}=\frac{{25}}{{8}}$$
We know that in two triangles $$\displaystyle\triangle{A}{B}{C}{\quad\text{and}\quad}\triangle{P}{Q}{R}$$ both will be similar if
$$\displaystyle\frac{{{A}{B}}}{{{P}{Q}}}=\frac{{{B}{C}}}{{{Q}{R}}}=\frac{{{C}{A}}}{{{R}{P}}}$$
Step 2
Similarly, in $$\displaystyle\triangle{D}{E}{F}{\quad\text{and}\quad}\triangle{E}{G}{F}$$
$$\displaystyle\frac{{{D}{E}}}{{{E}{G}}}=\frac{{5}}{{{\left(\frac{{25}}{{8}}\right)}}}$$
$$\displaystyle=\frac{{{5}\times{8}}}{{25}}$$
$$\displaystyle=\frac{{8}}{{5}}$$
$$\displaystyle\frac{{{E}{F}}}{{{G}{F}}}=\frac{{4}}{{{\left(\frac{{5}}{{2}}\right)}}}$$
$$\displaystyle=\frac{{{4}\times{2}}}{{5}}$$
$$\displaystyle=\frac{{8}}{{5}}$$
$$\displaystyle\frac{{{D}{F}}}{{{E}{F}}}=\frac{{{\left(\frac{{32}}{{5}}\right)}}}{{4}}$$
$$\displaystyle=\frac{{8}}{{5}}$$
Step 3
So, in $$\displaystyle\triangle{D}{E}{F}{\quad\text{and}\quad}\triangle{E}{G}{F}$$
$$\displaystyle\frac{{{D}{E}}}{{{E}{G}}}=\frac{{{E}{F}}}{{{G}{F}}}=\frac{{{D}{F}}}{{{E}{F}}}=\frac{{8}}{{5}}$$
So, by equation(1) both triangles are similar by SSS criteria
Hence, $$\displaystyle\triangle{D}{E}{F}{\quad\text{and}\quad}\triangle{E}{G}{F}$$ are similar by SSS criteria.