# Triangle A has sides of lengths 39 , 45 , and 27 . Triangle B is similar to triangle A and has a side of length 3 . What are the possible lengths of the other two sides of triangle B?

Jul 14, 2017

$\left(3 , \frac{45}{13} , \frac{27}{13}\right) , \left(\frac{13}{5} , 3 , \frac{9}{5}\right) , \left(\frac{13}{3} , 5 , 3\right)$

#### Explanation:

Since triangle B has 3 sides, anyone of them could be of length 3 and so there are 3 different possibilities.

Since the triangles are similar then the ratios of corresponding sides are equal.

Label the 3 sides of triangle B, a, b and c corresponding to the sides 39, 45 and 27 in triangle A.

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$\text{if a = 3 then ratio of corresponding sides } = \frac{3}{39} = \frac{1}{13}$

$\Rightarrow b = 45 \times \frac{1}{13} = \frac{45}{13} \text{ and } c = 27 \times \frac{1}{13} = \frac{27}{13}$

$\text{the 3 sides of B} = \left(3 , \textcolor{red}{\frac{45}{13}} , \textcolor{red}{\frac{27}{13}}\right)$
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$\text{if b = 3 then ratio of corresponding sides } = \frac{3}{45} = \frac{1}{15}$

$\Rightarrow a = 39 \times \frac{1}{15} = \frac{13}{5} \text{ and } c = 27 \times \frac{1}{15} = \frac{9}{5}$

$\text{the 3 sides of B } = \left(\textcolor{red}{\frac{13}{5}} , 3 , \textcolor{red}{\frac{9}{5}}\right)$
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$\text{if c = 3 then ratio of corresponding sides } = \frac{3}{27} = \frac{1}{9}$

$\Rightarrow a = 39 \times \frac{1}{9} = \frac{13}{3} \text{ and } b = 45 \times \frac{1}{9} = 5$

$\text{the 3 sides of B } = \left(\textcolor{red}{\frac{13}{3}} , \textcolor{red}{5} , 3\right)$

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