Question

Triangle A has an area of `84` and two sides of lengths `18` and `15`. Triangle B is similar to triangle A and has a side of length `5`. What are the maximum and minimum possible areas of triangle B?

Answer 1

`16.634` and `2.161`

Explanation:

Let me say something obvious:
- the smallest possible similar triangle has the minimum area and the largest possible similar triangle has the maximum area.

Suppose that in triangle A, the unknown side is `a`.
Suppose that in triangle B, the known side is `d`.

The smallest triangle B occurs when side `d` is proportional to the largest side of triangle A (then the other sides will be smaller than `d`). The largest triangle B occurs when side `d` is proportional to the smallest side of triangle A (then the other sides will be larger than `d`).
(The triangle may also be an isosceles one, in which case there will two big congruent sides or two small congruent sides).

Basically is a matter of knowing the length of side `a`.

In the Heron's formula for the area of the triangle:

`S=sqrt(s(s-a)(s-b)(s-c))`
`s=(a+b+c)/2=(a+18+15)/2=(a+33)/2`
`84=sqrt((a+33)/2*((a+33)/2-a)((a+33)/2-18)((a+33)/2-15)`
`7056=(a+33)/2*(-a+33)/2*(a-3)/2*(a+3)/2`
`112896=(-a^2+1089)(a^2-9)`
`112896=-a^4+9a^2+1089a^2-9801`
`a^4-1098a^2+122697=0`
`-> Delta=1,205,604-490,788=714,816`
`-> sqrt(Delta)=845.468`
`a^2=(1098+-845.468)/2`
`->a_1^2=971.733` => `a_1=31.173`
`->a_2^2=126.666` => `a_2=11.236`

As we can see triangle A can have 2 different shapes, one in which side `a` is the largest one and other in which side `a` is the smallest one.

If two triangles are similar their sides are directly proportional (`s"'"=k*s`) and so are their heights (`h"'"=k*h`), then:
`(S"'")/S=((b"'"*h"'")/cancel(2))/((b*h)/cancel(2))=((k*cancel(b))(k*cancel(h)))/(cancel(b)*cancel(h))=k^2`
`-> S"'"=k^2*S`, where k is the ratio between corresponding sides

For `a=11.236`

`S"'"=(5/11.236)^2*84=16.634` (maximum area)

For `a=31.173`

`S"'"=(5/31.173)^2*84=2.161` (minimum area)