Question

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?

Answer 1

Maximum possible area of triangle B `A_(Bmax) = color(green)(205.5919)`

Minimm possible area of triangle B `A_(Bmin) = color(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 `color(brown)(a/b)` then the areas will be in the ratio `color(blue)(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_(Bmax) = A_A * (12/4.1)^2 = 24 * (12/4.1)^2 = color(green)(205.5919)`

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

`A_(Bmin) = A_A * (12/19.9)^2 = 24 * (12/19.9)^2 = color(red)(8.7271)`