# Triangle A has an area of 4  and two sides of lengths 9  and 3 . Triangle B is similar to triangle A and has a side with a length of 32 . What are the maximum and minimum possible areas of triangle B?

Feb 18, 2018

A_(Bmax) = color(green)( 110.0779

A_(Bmin) = color(red)(28.9245

#### Explanation: Given : Delta A (PQR) p = 9, q = 3, A_A = 4, z = 32

To find ${A}_{B r a x \to n} , {A}_{B \min}$

Let x,y,z be the sides of second $\Delta B$

$r < \left(9 + 3\right) , > \left(9 - 3\right)$ as sum of the two sides greater than the third side.

Hence, $r = 6.1 .$, for maximum area of $\Delta B$ and ${r}_{\min} = 11.9$ for minimum area of $\Delta B$, r being rounded for one decimal.

We know, $\frac{p}{x} = \frac{q}{y} , \frac{r}{z}$ since the two triangles are similar.

Then he areas will be proportional to the square of the sides.

$\therefore {A}_{B} = {A}_{A} \cdot {\left(\frac{z}{r}\right)}^{2}$

A_(Bmax) = 4 * (32/6.1)^2 =color(green)( 110.0779

A_(Bmin) = 4 * (32/11.9)^2 = color(red)(28.9245#