# Triangle A has sides of lengths 1 3 ,1 4, and 1 8. 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?

May 25, 2018

$\frac{56}{13} \mathmr{and} \frac{72}{13} , \frac{26}{7} \mathmr{and} \frac{36}{7} , \mathmr{and} \frac{26}{9} \mathmr{and} \frac{28}{9}$

#### Explanation:

Since the triangles are similar, that means that the side lengths have the same ratio, i.e. we can multiply all of the lengths and get another. For example, an equilateral triangle has side lengths (1, 1, 1) and a similar triangle might have lengths (2, 2, 2) or (78, 78, 78), or something similar. An isosceles triangle may have (3, 3, 2) so a similar may have (6, 6, 4) or (12, 12, 8).

So here we start with (13, 14, 18) and we have three possibilities:
(4, ?, ?) , (?, 4, ?), or (?, ?, 4). Therefore, we ask what the ratios are.

If the first, that means the lengths are multiplied by $\frac{4}{13}$.
If the second, that means the lengths are multiplied by $\frac{4}{14} = \frac{2}{7}$
If the third, that means the lengths are multiplied by $\frac{4}{18} = \frac{2}{9}$

So we therefore have potential values
$\frac{4}{13} \cdot \left(13 , 14 , 18\right) = \left(4 , \frac{56}{13} , \frac{72}{13}\right)$
$\frac{2}{7} \cdot \left(13 , 14 , 18\right) = \left(\frac{26}{7} , 4 , \frac{36}{7}\right)$
$\frac{2}{9} \cdot \left(13 , 14 , 18\right) = \left(\frac{26}{9} , \frac{28}{9} , 4\right)$