# Triangle A has an area of 24  and two sides of lengths 8  and 12 . Triangle B is similar to triangle A and has a side with a length of 12 . What are the maximum and minimum possible areas of triangle B?

Jan 28, 2018

Maximum possible area of triangle B ${A}_{B \max} = \textcolor{g r e e n}{205.5919}$

Minimm possible area of triangle B ${A}_{B \min} = \textcolor{red}{8.7271}$

#### Explanation:

Third side of Triangle A can have values between 4 & 20 only by applying the condition that

Sum of the two sides of a triangle must be greater than the third side.

Let the values be 4.1 & 19.9. (corrected to one decimal point.

if sides are in the ratio $\textcolor{b r o w n}{\frac{a}{b}}$ then the areas will be in the ratio $\textcolor{b l u e}{{a}^{2} / {b}^{2}}$

Case - Max : When side 12 of corresponds to 4.1 of A, we get the maximum area of triangle B.

${A}_{B \max} = {A}_{A} \cdot {\left(\frac{12}{4.1}\right)}^{2} = 24 \cdot {\left(\frac{12}{4.1}\right)}^{2} = \textcolor{g r e e n}{205.5919}$

Case - Min : When side 12 of corresponds to 19.9 of A, we get the minimum area of triangle B.

${A}_{B \min} = {A}_{A} \cdot {\left(\frac{12}{19.9}\right)}^{2} = 24 \cdot {\left(\frac{12}{19.9}\right)}^{2} = \textcolor{red}{8.7271}$