# Triangle A has sides of lengths 28 , 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?

Nov 11, 2016

$\left(8 , \frac{72}{7} , \frac{48}{7}\right) , \left(\frac{56}{9} , 8 , \frac{16}{3}\right) , \left(\frac{28}{3} , 12 , 8\right)$

#### Explanation:

Anyone of the 3 sides of triangle B could be of length 8, hence there are 3 different possibilities for the sides of B.

Since the triangles are similar then the $\textcolor{b l u e}{\text{ratios of corresponding sides are equal}}$

Label the 3 sides of triangle B, a, b and c to correspond with the sides 28, 36 and 24 in triangle A.
$\textcolor{b l u e}{\text{-----------------------------------------------------}}$
If side a = 8 then ratio of corresponding sides $= \frac{8}{28} = \frac{2}{7}$

and side b $= 36 \times \frac{2}{7} = \frac{72}{7} , \text{ side c} = 24 \times \frac{2}{7} = \frac{48}{7}$

The 3 sides of B would be $\left(8 , \textcolor{red}{\frac{72}{7}} , \textcolor{red}{\frac{48}{7}}\right)$
$\textcolor{b l u e}{\text{-----------------------------------------------------------}}$
If side b = 8 then ratio of corresponding sides $= \frac{8}{36} = \frac{2}{9}$

and side a $= 28 \times \frac{2}{9} = \frac{56}{9} , c = 24 \times \frac{2}{9} = \frac{48}{9}$

The 3 sides of B would be $\left(\textcolor{red}{\frac{56}{9}} , 8 , \textcolor{red}{\frac{48}{9}}\right)$
$\textcolor{b l u e}{\text{-----------------------------------------------------------------}}$
If side c = 8 then ratio of corresponding sides $= \frac{8}{24} = \frac{1}{3}$

and side a $= 28 \times \frac{1}{3} = \frac{28}{3} , b = 36 \times \frac{1}{3} = 12$

The 3 sides of B would be $\left(\textcolor{red}{\frac{28}{3}} , \textcolor{red}{12} , 8\right)$
$\textcolor{b l u e}{\text{----------------------------------------------------------------}}$