Question

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?

Answer 1

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

`12/B_1 = 16/B_2 = 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`

`12/color(red)(16) = 3/4`
`3/4 = 16/B_2 => B_2=21.333`
`3/4 = 18/B_3 => B_3=24`

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

If `B_2=16`

`16/color(red)(16) = 1 =>` 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`

`18/color(red)(16) = 9/8`
`9/8 = 12/B_1 => B_1 = 10.667`
`9/8 = 16/B_2 => B_2 = 14.222`

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