Question

Triangle A has sides of lengths `60`, `42`, and `60`. Triangle B is similar to triangle A and has a side of length `7`. What are the possible lengths of the other two sides of triangle B?

Answer 1

`10 and 4.9`

Explanation:

`color(white)(WWWW)color (black)Delta B"color(white)(WWWWWWWWWWWWWW)color (black)Delta A`

Let two triangles `A and B` be similar. `DeltaA` is `OPQ` and has sides `60,42 and 60`. Since two sides are equal to each other it is an isosceles triangle.
and `DeltaB` is `LMN` has one side`=7`.
By properties of Similar Triangles

1. Corresponding angles are equal and
2. Corresponding sides are all in the same proportion.

It follows that `DeltaB` must also be an isosceles triangle.

There are two possibilities
(a) Base of `DeltaB` is `=7`.
From proportionality
`"Base"_A/"Base"_B="Leg"_A/"Leg"_B` .....(1)
Inserting given values
`42/7=60/"Leg"_B`
`=>"Leg"_B=60xx7/42`
`=>"Leg"_B=10`

(b) Leg of `DeltaB` is `=7`.
From equation (1)
`42/"Base"_B=60/7`
`"Base"_B=42xx7/60`
`"Base"_B=4.9`