# Triangle A has sides of lengths 51 , 45 , and 54 . 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 24, 2018

See below.

#### Explanation:

For similar triangles we have:

$\frac{A}{B} = \frac{A '}{B '} \textcolor{w h i t e}{888888}$ $\frac{A}{C} = \frac{A '}{C '}$ etc.

Let $A = 51 , B = 45 , C = 54$

Let $A ' = 3$

$\frac{A}{B} = \frac{51}{45} = \frac{3}{B '} \implies B ' = \frac{45}{17}$

$\frac{A}{C} = \frac{51}{54} = \frac{3}{C '} \implies C ' = \frac{54}{17}$

1st set of possible sides: $\left\{3 , \frac{45}{17} , \frac{54}{17}\right\}$

Let $B ' = 3$

$\frac{A}{B} = \frac{51}{45} = \frac{A '}{3} \implies A ' = \frac{17}{5}$

$\frac{B}{C} = \frac{45}{54} = \frac{3}{C '} \implies C ' = \frac{18}{5}$

2nd set of possible sides $\left\{\frac{17}{5} , 3 , \frac{18}{5}\right\}$

Let $C ' = 3$

$\frac{A}{C} = \frac{51}{54} = \frac{A '}{3} \implies A ' = \frac{17}{6}$

$\frac{B}{C} = \frac{45}{54} = \frac{B '}{3} \implies B ' = \frac{5}{2}$

3rd set of possible sides $\left\{\frac{17}{6} , \frac{5}{2} , 3\right\}$