# Triangles ABC and DEF are similar triangles. Use this fact

Triangles ABC and DEF are similar triangles. Use this fact to solve the exercise. Round to the nearest tenth.
Find the perimeter of triangle ABC.

• Questions are typically answered in as fast as 30 minutes

### Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Befory

Concept used:
(1) Pythagoras theorem: In a right angle triangle
$$\displaystyle{\left({H}{y}{p}{o}{t}{e}nu{s}{e}\right)}^{{{2}}}={\left({P}{e}{r}{p}{e}{n}{d}{i}{c}ul{{a}}{r}\right)}^{{{2}}}+{\left({B}{a}{s}{e}\right)}^{{{2}}}$$
(2) If two triangles are similar, their perimeter have the exact same ratio.
In right angle triangle DEF
$$\displaystyle{\left({H}{y}{p}{o}{t}{e}nu{s}{e}\right)}^{{{2}}}={\left({P}{e}{r}{p}{e}{n}{d}{i}{c}ul{{a}}{r}\right)}^{{{2}}}+{\left({B}{a}{s}{e}\right)}^{{{2}}}$$
$$\displaystyle{\left({D}{E}\right)}^{{{2}}}={\left({E}{F}\right)}^{{{2}}}+{\left({F}{D}\right)}^{{{2}}}$$
$$\displaystyle{\left({D}{E}\right)}={\left({15}\right)}^{{{2}}}+{\left({6}\right)}^{{{2}}}$$
$$\displaystyle={225}+{1296}$$
$$\displaystyle={1521}$$
$$\displaystyle{D}{E}=\pm\sqrt{{{1521}}}$$
$$\displaystyle=\pm{139}$$
Side can't be negative so DE =39
By similar triangle property:
$$\displaystyle{\frac{{{p}{e}{r}{i}{m}{e}{t}{e}{r}\ {o}{f}\ \triangle{D}{E}{F}}}{{{p}{e}{r}{i}{m}{e}{t}{e}{r}\ {o}{f}\ \triangle{A}{B}{C}}}}={\frac{{{D}{F}}}{{{A}{C}}}}={\frac{{{D}{E}}}{{{A}{B}}}}={\frac{{{E}{F}}}{{{B}{C}}}}$$

$$\displaystyle{p}{e}{r}{i}{m}{e}{t}{e}{r}\ {o}{f}\ \triangle{D}{E}{F}={39}+{36}+{15}={90}$$

$$\displaystyle{\frac{{{p}{e}{r}{i}{m}{e}{t}{e}{r}\ {o}{f}\ \triangle{D}{E}{F}}}{{{p}{e}{r}{i}{m}{e}{t}{e}{r}\ {o}{f}\ \triangle{A}{B}{C}}}}={\frac{{{D}{F}}}{{{A}{C}}}}$$

$$\displaystyle{\frac{{{90}}}{{{p}{e}{r}{i}{m}{e}{t}{e}{r}\ {o}{f}\ \triangle{A}{B}{C}}}}={\frac{{{36}}}{{{12}}}}$$

$$\displaystyle{p}{e}{r}{i}{m}{e}{t}{e}{r}\ {o}{f}\ \triangle{A}{B}{C}={\frac{{{90}\times{12}}}{{{36}}}}$$

$$\displaystyle{p}{e}{r}{i}{m}{e}{t}{e}{r}\ {o}{f}\ \triangle{A}{B}{C}={30}$$

###### Not exactly what youâ€™re looking for?

• Questions are typically answered in as fast as 30 minutes