# Triangle A has sides of lengths 32 , 36 , and 24 . Triangle B is similar to triangle A and has a side of length 8 . What are the possible lengths of the other two sides of triangle B?

Feb 15, 2016

The other two sides of B are
$\textcolor{w h i t e}{\text{XXX}} \left\{9 , 6\right\}$
or
$\textcolor{w h i t e}{\text{XXX}} \left\{7 \frac{1}{9} , 5 \frac{1}{3}\right\}$
or
$\textcolor{w h i t e}{\text{XXX}} \left\{12 , 10 \frac{2}{3}\right\}$

#### Explanation:

Denote the given side of B as ${B}_{1} = 8$
and
the sides of A as ${A}_{1} , {A}_{2} , {A}_{3}$ such that ${A}_{1}$ corresponds to ${B}_{1}$

Given: $\left\{{A}_{1} , {A}_{2} , {A}_{3}\right\} = \left\{32 , 36 , 24\right\}$

Case 1 : ${A}_{1} = 32$
$\textcolor{w h i t e}{\text{XXX}}$The ratio of ${A}_{1} : {B}_{1} = 32 : 8 = 4 : 1$
$\textcolor{w h i t e}{\text{XXX}}$All corresponding sides must be in this ratio
$\textcolor{w h i t e}{\text{XXX}} \rightarrow {B}_{x} = {A}_{x} / 4$
$\textcolor{w h i t e}{\text{XXX}}$So the other two sides of B must have lengths
$\textcolor{w h i t e}{\text{XXX}} \frac{36}{4} = 9 \mathmr{and} \frac{24}{4} = 6$

Case 1 : ${A}_{1} = 36$
$\textcolor{w h i t e}{\text{XXX}}$The ratio of ${A}_{1} : {B}_{1} = 36 : 8 = 4.5 : 1$
$\textcolor{w h i t e}{\text{XXX}}$All corresponding sides must be in this ratio
$\textcolor{w h i t e}{\text{XXX}} \rightarrow {B}_{x} = {A}_{x} / 4.5$
$\textcolor{w h i t e}{\text{XXX}}$So the other two sides of B must have lengths
$\textcolor{w h i t e}{\text{XXX}} \frac{32}{4.5} = 7 \frac{1}{9} \mathmr{and} \frac{24}{4.5} = 5 \frac{1}{3}$

Case 1 : ${A}_{1} = 24$
$\textcolor{w h i t e}{\text{XXX}}$The ratio of ${A}_{1} : {B}_{1} = 24 : 8 = 3$
$\textcolor{w h i t e}{\text{XXX}}$All corresponding sides must be in this ratio
$\textcolor{w h i t e}{\text{XXX}} \rightarrow {B}_{x} = {A}_{x} / 3$
$\textcolor{w h i t e}{\text{XXX}}$So the other two sides of B must have lengths
$\textcolor{w h i t e}{\text{XXX}} \frac{36}{3} = 12 \mathmr{and} \frac{32}{3} = 10 \frac{2}{3}$