# Triangle A has sides of lengths 24 , 16 , and 18 . Triangle B is similar to triangle A and has a side with a length of 16 . What are the possible lengths of the other two sides of triangle B?

Sep 23, 2016

$\left(16 , \frac{32}{3} , 12\right) , \left(24 , 16 , 18\right) , \left(\frac{64}{3} , \frac{128}{9} , 16\right)$

#### Explanation:

Anyone of the 3 sides of triangle B could be of length 16 hence there are 3 different possibilities for the sides of B.
Since the triangles are similar then the $\textcolor{b l u e}{\text{ratios of corresponding sides are equal}}$

Name the 3 sides of triangle B- a , b and c to correspond with the sides- 24 , 16 and 18 in triangle A.
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If side a = 16 then ratio of corresponding sides $= \frac{16}{24} = \frac{2}{3}$

and side b $= 16 \times \frac{2}{3} = \frac{32}{3} , \text{ side c} = 18 \times \frac{2}{3} = 12$

The 3 sides of B would be $\left(16 , \textcolor{red}{\frac{32}{3}} , \textcolor{red}{12}\right)$
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If side b = 16 then ratio of corresponding sides $= \frac{16}{16} = 1$
and side a $= 24 \text{, side c} = 18$

The 3 sides of B would be $\left(\textcolor{red}{24} , 16 , \textcolor{red}{18}\right)$
$\textcolor{b l u e}{\text{-----------------------------------------------------------------}}$

If side c = 16 then ratio of corresponding sides $= \frac{16}{18} = \frac{8}{9}$

and side a $= 24 \times \frac{8}{9} = \frac{64}{3} , \text{ side b} = 16 \times \frac{8}{9} = \frac{128}{9}$

The 3 sides of B would be $\left(\textcolor{red}{\frac{64}{3}} , \textcolor{red}{\frac{128}{9}} , 16\right)$
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