Question

What is the radius of the largest circle that can be inscribed in a triangle with side lengths 13, 14 and 15?

Answer 1

The center of the inscribed circle is at the intersection of the bisectors of the angles. The radius is the distance from that center to each side perpendicular to that side.

Answer 2

Use Heron's formula to find the area A.

Then A = rs
where s is the semi-perimeter.

Answer 3

[AI-generated answer — Explain Bot]

To find the radius of the largest circle that can be inscribed in a triangle, we need to use the formula known as the inradius formula. This formula relates the inradius of the triangle to its side lengths.

The inradius (r) is defined as the radius of the largest circle that can be inscribed in a triangle. It can be calculated using the formula:

r = Area / (s)

Where "s" is the semi-perimeter of the triangle and is calculated by:

s = (a + b + c) / 2

And "Area" is the area of the triangle, which can be calculated using Heron's formula:

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

In this case, the side lengths of the triangle are given as 13, 14, and 15.

First, calculate the semi-perimeter (s) by adding the three side lengths and dividing by 2:

s = (13 + 14 + 15) / 2 = 42 / 2 = 21

Next, use Heron's formula to calculate the area of the triangle:

Area = sqrt(s * (s - a) * (s - b) * (s - c))
= sqrt(21 * (21 - 13) * (21 - 14) * (21 - 15))
= sqrt(21 * 8 * 7 * 6)
= sqrt(21 * 8 * 7 * 6)
= sqrt(2112)
≈ 45.99

Finally, substitute the values of the area and the semi-perimeter into the formula for the inradius:

r = Area / s
r = 45.99 / 21 ≈ 2.19

Therefore, the radius of the largest circle that can be inscribed in the triangle with side lengths 13, 14, and 15 is approximately 2.19 units.