The sides of the triangle are 8 10 14. Find the radius of the circle inscribed in this triangle.

We know from the condition that the triangle has side lengths of 8, 10 and 14. In order to find the radius of the inscribed circle, we apply the formula for calculating the area:

S = p * r;

Let us express the radius of the inscribed circle from the formula:

r = S / p.

To calculate the area, we will use Heron’s formula. Let’s remember the formula first:

S = √p (p – a) (p – b) (p – c), where p = (a + b + c) / 2.

It remains to substitute the values into the formula and perform calculations:

p = (a + b + c) / 2 = (8 + 10 + 14) / 2 = 32/2 = 16;

S = √16 (16 – 8) (16 – 10) (16 – 14) = √16 * 8 * 6 * 2 = 16√6.

r = 16√6 / 16 = √6.