# Triangle A has sides of lengths 54 , 44 , and 64 . 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?

Apr 3, 2016

$\left(8 , \frac{176}{27} , \frac{256}{27}\right) , \left(\frac{108}{11} , 8 , \frac{128}{11}\right) , \left(\frac{27}{4} , \frac{11}{2} , 8\right)$

#### Explanation:

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 54 , 44 and 64 in triangle A.
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If side a = 8 then ratio of corresponding sides = $\frac{8}{54} = \frac{4}{27}$

Hence b = $44 \times \frac{4}{27} = \frac{176}{27} \text{ and } c = 64 \times \frac{4}{27} = \frac{256}{27}$

The 3 sides in B $= \left(8 , \frac{176}{27} , \frac{256}{27}\right)$
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If side b = 8 then ratio of corresponding sides$= \frac{8}{44} = \frac{2}{11}$

hence a = $54 \times \frac{2}{11} = \frac{108}{11} \text{ and } c = 64 \times \frac{2}{11} = \frac{128}{11}$

The 3 sides in B = $\left(\frac{108}{11} , 8 , \frac{128}{11}\right)$
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If side c = 8 then ratio of corresponding sides $= \frac{8}{64} = \frac{1}{8}$

hence a $= 54 \times \frac{1}{8} = \frac{27}{4} \text{ and } b = 44 \times \frac{1}{8} = \frac{11}{2}$

The 3 sides in B =$\left(\frac{27}{4} , \frac{11}{2} , 8\right)$
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