Question

Triangle A has an area of `6` and two sides of lengths `5` and `8`. Triangle B is similar to triangle A and has a side with a length of `19`. What are the maximum and minimum possible areas of triangle B?

Answer 1

Case 1. Maximum Area of `Delta B` = `color(green)(225.3902)` sq. units

Case 2. Minimum Area of `Delta B` = `color(red)(13.016)` sq. units

Explanation:

`Delta A = 6, p = 5, q = 8, x = 19`

Side r can have values between `q-p) , (q + p)`

`r > (8-5), < (8 + 5)`

`r>3 <13`

`r_(min) = 3.1, r_(max) = 12.9`, rounded to one decemal.

Case 1. Maximum Area of `Delta B`

x should correspond to least side of A, viz. r = 3.1 to get minimum area of B.

`Delta B / Delta A = (x / r)^2`

`Delta B = 6 * (19 / 3.1)^2 = color(green)(225.3902)` sq. units

Case 2. Minimum Area of `Delta B`

x should correspond to longest side of A, viz. r = 12.9 to get maximum area of B.

`Delta B / Delta A = (x / r)^2`

`Delta B = 6 * (19 / 12.9)^2 = color(red)(13.016)` sq. units