# Triangle A has an area of 24  and two sides of lengths 9  and 6 . Triangle B is similar to triangle A and has a side of length 9 . What are the maximum and minimum possible areas of triangle B?

Feb 1, 2018

${B}_{\max} = \textcolor{g r e e n}{\frac{A \cdot {z}^{2}}{r} ^ 2 \approx 202.2893}$

${B}_{\min} = \textcolor{red}{\frac{A \cdot {z}^{2}}{r} \approx 8.7564}$

#### Explanation: Triangle A has sides p, q, r and area A

Triangle B has sides x, y, x with area B.

p = 9, q = 6.

r can have values between $\left(9 - 6\right) \to \left(9 + 6\right)$ using the property, sum of the two sides greater than the third side of a triangle.

Min. value 3.1 and max. value 14.9 ( taking one decimal correction).

Case 1 : r = 3.1 and z = 9

We know, $\frac{B}{A} = {\left(\frac{z}{r}\right)}^{2}$

B_max = color(green((A * z^2) / r^2) = (24 * 9^2) / 3.1^2 ~~ color(green)(202.2893)

Case 2 : r = 14.9 and z = 9

We know, $\frac{B}{A} = {\left(\frac{z}{r}\right)}^{2}$

${B}_{\min} = \textcolor{red}{\frac{A \cdot {z}^{2}}{r} ^ 2} = \frac{24 \cdot {9}^{2}}{14.9} ^ 2 \approx \textcolor{red}{8.7564}$