Calculate the area of a circle inscribed in a triangle, the sides of which are 10cm, 24cm, 26cm.

Let’s designate the sides of the triangle ABC:

a = 10 cm, b = 24 cm, c = 26 cm.

Note that by the cosine theorem:

c ^ 2 = a ^ 2 + b ^ 2 – 2 * a * b * cos (C),

26 ^ 2 = 10 ^ 2 + 24 ^ 2 – 2 * 10 * 24 * cos (C),

676 = 576 + 100 – 2 * 10 * 24 * cos (C),

cos (C) = 0, C = 90 °.

Hence, triangle ABC is rectangular and its area is S:

S = 1/2 * a * b = 1/2 * 10 * 24 = 120.

And its perimeter P:

P = a + b + c = 10 + 24 + 26 = 60.

If r is the radius of the inscribed circle, then:

S = 1/2 * P * r = 30 * r = 120, r = 4.

Therefore, the area of the inscribed circle s:

s = pi * r ^ 2 = pi * 4 ^ 2 = 16 * pi.

Answer: 16 * pi.