Question

Triangle A has an area of `3` and two sides of lengths `3` and `6`. Triangle B is similar to triangle A and has a side with a length of `11`. What are the maximum and minimum possible areas of triangle B?

Answer 1

The triangle inequality states that the sum of any two sides of a triangle MUST be greater than the 3rd side. That implies the missing side of triangle A must be greater than 3!

Explanation:

Using the triangle inequality ...

`x+3>6`
`x>3`

So, the missing side of triangle A must fall between 3 and 6.

This means 3 is the shortest side and 6 is the longest side of triangle A.

Since area is proportional to the square of the ratio of the similar sides ...

minimum area `= (11/6)^2xx3=121/12~~10.1`

maximum area `=(11/3)^2xx3=121/3~~40.3`

Hope that helped

P.S. - If you really want to know the length of the missing 3rd side of triangle A, you can use Heron's area formula and determine that the length is `~~3.325`. I'll leave that proof to you :)