# Triangle A has sides of lengths 54 , 44 , and 64 . Triangle B is similar to triangle A and has a side of length 4 . What are the possible lengths of the other two sides of triangle B?

Feb 7, 2017

$< 4 , 3 \frac{7}{27} , 4 \frac{20}{27} >$, $< 4 \frac{10}{11} , 4 , 5 \frac{9}{11} >$ and$< 3 \frac{3}{8} , 2 \frac{3}{4} , 4 >$

#### Explanation:

Let $\left(4 , a , b\right)$ are the lengths of Triangle B..

A. Comparing 4 and 54 from Triangle A,

$\frac{b}{44} = \frac{4}{54}$, $b = \frac{2}{27} \cdot 44 = 3 \frac{7}{27}$

$\frac{c}{64} = \frac{4}{54}$, $c = \frac{2}{27} \cdot 64 = 4 \frac{20}{27}$

The length of sides for Triangle B is$< 4 , 3 \frac{7}{27} , 4 \frac{20}{27} >$

B. Comparing 4 and 44 from Triangle A,

$\frac{b}{54} = \frac{4}{44}$, $b = \frac{1}{11} \cdot 54 = 4 \frac{10}{11}$

$\frac{c}{64} = \frac{4}{44}$, $c = \frac{1}{11} \cdot 64 = 5 \frac{9}{11}$

The length of sides for Triangle B is$< 4 \frac{10}{11} , 4 , 5 \frac{9}{11} >$

Comparing 4 and 64 from Triangle A,

$\frac{b}{54} = \frac{4}{64}$,$b = \frac{1}{16} \cdot 54 = 3 \frac{3}{8}$

$\frac{c}{44} = \frac{4}{64}$, $c = \frac{1}{16} \cdot 44 = 2 \frac{3}{4}$
The length of sides for Triangle B is$< 3 \frac{3}{8} , 2 \frac{3}{4} , 4 >$

Therefore the possible sides for Triangle B are

$< 4 , 3 \frac{7}{27} , 4 \frac{20}{27} >$, $< 4 \frac{10}{11} , 4 , 5 \frac{9}{11} >$ and$< 3 \frac{3}{8} , 2 \frac{3}{4} , 4 >$