The sides of a triangle are in the ratio 3:5:6. If its perimeter is 56 cm, use the Heroes formula to find its area

First, let's find the lengths of the sides of the triangle. If the sides are in the ratio 3:5:6, let's denote the sides as 3x, 5x, and 6x. The perimeter of the triangle is the sum of its sides, so:

3x + 5x + 6x = 56 14x = 56 x = 56 / 14 x = 4

So, the sides of the triangle are 3x = 12 cm, 5x = 20 cm, and 6x = 24 cm.

Next, let's use Heron's formula to find the area of the triangle. Heron's formula is used to calculate the area of a triangle with the lengths of all three sides. The formula is:

Area = sqrt[s(s - a)(s - b)(s - c)]

where a, b, and c are the sides of the triangle, and s is the semi-perimeter of the triangle, calculated as (a + b + c) / 2.

So, s = (12 + 20 + 24) / 2 = 28 cm.

Now, let's substitute s, a, b, and c into Heron's formula:

Area = sqrt[28(28 - 12)(28 - 20)(28 - 24)] Area = sqrt[28 * 16 * 8 * 4] Area = sqrt[112896] Area = 336 cm^2

So, the area of the triangle is 336 square cm.