Question

Find the side of an equilateral triangle whose area is equal to the area of that triangle whose sides are 12 cm, 15 cm and 21 cm.

Answer 1

Given: The triangle with sides 12 cm, 15 cm and 21 cm; find the side a of an equilateral triangle whose area equals the area of that triangle.

Step-wise calculation:

1. Compute semiperimeter:

`s = (12 + 15 + 21)/2`

= `48/2`

= 24 cm

2. Area by Heron’s formula:

Area = `sqrt(s(s - a)(s - b)(s - c))`

= `sqrt(24(24 - 12)(24 - 15)(24 - 21))`

= `sqrt(24 xx 12 xx 9 xx 3)`

= `sqrt(7776)`

= `36sqrt(6)  cm^2`

3. Equate to area of an equilateral triangle (side = x):

`(sqrt(3)/4) x^2 = 36sqrt(6)`

4. Solve for x²:

`x^2 = (36sqrt(6)) xx (4/sqrt(3))`

= `144 xx sqrt(6/3)`

= `144 xx sqrt(2)`

5. Therefore, `x = sqrt(144sqrt(2))`

= `12 xx 2^(1/4) cm`

= 14.2705 cm

The side of the equilateral triangle is `x = 12 xx 2^(1/4) cm` (≈ 14.2705 cm).