Question

Triangle A has sides of lengths `51`, `45`, and `54`. 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

`105/17` and `126/17`; or
`119/15` and `42/5`; or
`119/18` and `35/6`

Explanation:

Two similar triangles have all of their side lengths in the same ratio. So, overall there are 3 possible `triangleB`s with a length of 7.

Case i) - the 51 length

So lets have the side length 51 go to 7. This is a scale factor of `7/51`. This means we multiply all of the sides by `7/51`

`51xx7/51=7`
`45xx7/51=315/51=105/17`
`54xx7/51=126/17`

So the lengths are (as fractions) `105/17` and `126/17`. You can give these as decimals, but generally fractions are better.

Case ii) - the 45 length

We do the same thing here. To get the side of 45 to 7, we multiply by `7/45`

`51xx7/45=119/15`
`45xx7/45=7`
`54xx7/45=42/5`

So the lengths are `119/15` and `42/5`

Case iii) - the 54 length

I'm hoping you know what to do by now. We multiply each length by `7/54`

`51xx7/54=119/18`
`45xx7/54=35/6`
`54xx7/54=7`

So the lengths are `119/18` and `35/6`

All of these triangles, although they have different side lengths, are all similar to triangle A, and all are answers.