The legs of a right-angled triangle are 30 and 40 cm, find the radius of the circle inscribed in this triangle.

The radius of a circle inscribed in a right-angled triangle is found by the formula:
r = (a + b – c) / 2,
where a and b are legs, c is hypotenuse.
By condition, a = 30 cm, b = 40 cm.
By the Pythagorean theorem, we find the hypotenuse with:
c ^ 2 = a ^ 2 + b ^ 2;
c = √ (a ^ 2 + b ^ 2);
c = √ (30 ^ 2 + 40 ^ 2) = √ (900 + 1600) = √2500 = 50 (cm).
Substitute the known values into the formula for the radius of the inscribed circle and find the length of the radius:
r = (30 + 40 – 50) / 2 = 20/2 = 10 (cm).
Answer: r = 10 cm.