# Triangle A has an area of 24  and two sides of lengths 8  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?

Jan 28, 2018

Case 1. ${A}_{B \max} \approx \textcolor{red}{11.9024}$

Case 2. ${A}_{B \min} \approx \textcolor{g r e e n}{1.1441}$

#### Explanation: Given Two sides of triangle A are 8, 15.

The third side should be $\textcolor{red}{> 7}$ and $\textcolor{g r e e n}{< 23}$, as sum of the two sides of a triangle should be greater than the third side.

Let the values of the third side be 7.1 , 22.9 ( Corrected upt one decimal point.

Case 1 : Third side = 7.1

Length of triangle B (5) corresponds to side 7.1 of triangle A to get the maximum possible area of triangle B

Then the areas will be proportionate by square of the sides.

${A}_{B \max} / {A}_{A} = {\left(\frac{5}{7.1}\right)}^{2}$

${A}_{B \max} = 24 \cdot {\left(\frac{5}{7.1}\right)}^{2} \approx \textcolor{red}{11.9024}$

Case 2 : Third side = 7.1

Length of triangle B (5) corresponds to side 22.9 of triangle A to get the minimum possible area of triangle B

${A}_{B \min} / {A}_{A} = {\left(\frac{5}{22.9}\right)}^{2}$

${A}_{B \min} = 24 \cdot {\left(\frac{5}{22.9}\right)}^{2} \approx \textcolor{g r e e n}{1.1441}$