# Triangle A has sides of lengths 15 , 12 , and 18 . 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?

May 17, 2016

$\left(3 , \frac{12}{5} , \frac{18}{5}\right) , \left(\frac{15}{4} , 3 , \frac{9}{2}\right) , \left(\frac{5}{2} , 2 , 3\right)$

#### 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.
Name the 3 sides of triangle B , a, b and c corresponding to the sides 15 , 12 and 18 in triangle A.
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If side a = 3 then the ratio of corresponding sides$= \frac{3}{15} = \frac{1}{5}$

hence b$= 12 \times \frac{1}{5} = \frac{12}{5} \text{ and } c = 18 \times \frac{1}{5} = \frac{18}{5}$

The 3 sides of B$= \left(3 , \frac{12}{5} , \frac{18}{5}\right)$
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If side b = 3 then the ratio of corresponding sides$= \frac{3}{12} = \frac{1}{4}$

hence a$= 15 \times \frac{1}{4} = \frac{15}{4} \text{ and } c = 18 \times \frac{1}{4} = \frac{9}{2}$

The 3 sides of B$= \left(\frac{15}{4} , 3 , \frac{9}{2}\right)$
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If side c = 3 then the ratio of corresponding sides$= \frac{3}{18} = \frac{1}{6}$

hence a$= 15 \times \frac{1}{6} = \frac{5}{2} \text{ and } b = 12 \times \frac{1}{6} = 2$

The 3 sides of B $= \left(\frac{5}{2} , 2 , 3\right)$
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