# 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?

Feb 15, 2016

$16.634$ 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:

$S = \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$
$s = \frac{a + b + c}{2} = \frac{a + 18 + 15}{2} = \frac{a + 33}{2}$
84=sqrt((a+33)/2*((a+33)/2-a)((a+33)/2-18)((a+33)/2-15)
$7056 = \frac{a + 33}{2} \cdot \frac{- a + 33}{2} \cdot \frac{a - 3}{2} \cdot \frac{a + 3}{2}$
$112896 = \left(- {a}^{2} + 1089\right) \left({a}^{2} - 9\right)$
$112896 = - {a}^{4} + 9 {a}^{2} + 1089 {a}^{2} - 9801$
${a}^{4} - 1098 {a}^{2} + 122697 = 0$
$\to \Delta = 1 , 205 , 604 - 490 , 788 = 714 , 816$
$\to \sqrt{\Delta} = 845.468$
${a}^{2} = \frac{1098 \pm 845.468}{2}$
$\to {a}_{1}^{2} = 971.733$ => ${a}_{1} = 31.173$
$\to {a}_{2}^{2} = 126.666$ => ${a}_{2} = 11.236$

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 ($s \text{'} = k \cdot s$) and so are their heights ($h \text{'} = k \cdot h$), then:
(S"'")/S=((b"'"*h"'")/cancel(2))/((b*h)/cancel(2))=((k*cancel(b))(k*cancel(h)))/(cancel(b)*cancel(h))=k^2
$\to S \text{'} = {k}^{2} \cdot S$, where k is the ratio between corresponding sides

For $a = 11.236$

$S \text{'} = {\left(\frac{5}{11.236}\right)}^{2} \cdot 84 = 16.634$ (maximum area)

For $a = 31.173$

$S \text{'} = {\left(\frac{5}{31.173}\right)}^{2} \cdot 84 = 2.161$ (minimum area)