In a triangle ABC, the sides are 2, 3, and 4. Find the radius of the circle inscribed in the triangle.

By Heron’s theorem, we determine the area of the triangle ABC.

The semi-perimeter of the ABC triangle is: p = (AB + BC + AC) / 2 = (2 + 3 + 4) / 2 = 9/2 = 4.5 cm.

Then Sаvs = √4.5 * (4.5 – 2) * (4.5 – 3) * (4.5 – 4) = √4.5 * 2.5 * 1.5 * 0.5 = √ 9 * 0.5 * 5 * 0.5 * 3 * 0.5 * 0.5 = 3 * 0.25 * √15 = 0.75 * √15 cm2.

Determine the radius of the inscribed circle.

R = S / p = 0.75 * √15 / 4.5 = √15 / 6 cm.

Answer: The radius of the inscribed circle is √15 / 6 cm.