# Triangle A has sides of lengths 12 , 16 , and 18 . Triangle B is similar to triangle A and has a side with a length of 16 . What are the possible lengths of the other two sides of triangle B?

Jan 23, 2017

There are 3 possible sets of lengths for Triangle B.

#### Explanation:

For triangles to be similar, all sides of Triangle A are in the same proportions to the corresponding sides in Triangle B.

If we call the lengths of the sides of each triangle {${A}_{1}$, ${A}_{2}$, and ${A}_{3}$} and {${B}_{1}$, ${B}_{2}$, and ${B}_{3}$}, we can say:

${A}_{1} / {B}_{1} = {A}_{2} / {B}_{2} = {A}_{3} / {B}_{3}$

or

$\frac{12}{B} _ 1 = \frac{16}{B} _ 2 = \frac{18}{B} _ 3$

The given information says that one of the sides of Triangle B is 16 but we don't know which side. It could be the shortest side (${B}_{1}$), the longest side (${B}_{3}$), or the "middle" side (${B}_{2}$) so we must consider all possibilities

If ${B}_{1} = 16$

$\frac{12}{\textcolor{red}{16}} = \frac{3}{4}$
$\frac{3}{4} = \frac{16}{B} _ 2 \implies {B}_{2} = 21.333$
$\frac{3}{4} = \frac{18}{B} _ 3 \implies {B}_{3} = 24$

{16, 21.333, 24} is one possibility for Triangle B

If ${B}_{2} = 16$

$\frac{16}{\textcolor{red}{16}} = 1 \implies$ This is a special case where Triangle B is exactly the same as Triangle A. The triangles are congruent.

{12, 16, 18} is one possibility for Triangle B.

If ${B}_{3} = 16$

$\frac{18}{\textcolor{red}{16}} = \frac{9}{8}$
$\frac{9}{8} = \frac{12}{B} _ 1 \implies {B}_{1} = 10.667$
$\frac{9}{8} = \frac{16}{B} _ 2 \implies {B}_{2} = 14.222$

{10.667, 14.222, 16} is one possibility for Triangle B.