# Triangle A has sides of lengths 2 ,3 , and 9 . Triangle B is similar to triangle A and has a side of length 1 . What are the possible lengths of the other two sides of triangle B?

Apr 12, 2016

$\left(1 , \frac{3}{2} , \frac{9}{2}\right) , \left(\frac{2}{3} , 1 , 3\right) , \left(\frac{2}{9} , \frac{1}{3} , 1\right)$

#### Explanation:

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

Name the 3 sides of triangle B , a , b and c , corresponding to the sides 2 , 3 and 9 in triangle A.
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If side a = 1 then ratio of corresponding sides $= \frac{1}{2}$
hence b = $3 \times \frac{1}{2} = \frac{3}{2} \text{ and } c = 9 \times \frac{1}{2} = \frac{9}{2}$
The 3 sides of B = $\left(1 , \frac{3}{2} , \frac{9}{2}\right)$
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If b = 1 then ratio of corresponding sides $= \frac{1}{3}$
hence a$= 2 \times \frac{1}{3} = \frac{2}{3} \text{ and } c = 9 \times \frac{1}{3} = 3$
The 3 sides of B = $\left(\frac{2}{3} , 1 , 3\right)$
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If c = 1 then ratio of corresponding sides$= \frac{1}{9}$
hence a $= 2 \times \frac{1}{9} = \frac{2}{9} \text{ and } b = 3 \times \frac{1}{9} = \frac{1}{3}$
The 3 sides of B = $\left(\frac{2}{9} , \frac{1}{3} , 1\right)$
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