Find the area of a triangle and the radii of a circle inscribed in a triangle and circumscribed

univerkov | July 4, 2021 | education

Find the area of a triangle and the radii of a circle inscribed in a triangle and circumscribed about a triangle, the sides of which are 11 cm, 25 cm, 30 cm.

We find the semiperimeter p of this triangle:

p = (11 + 25 + 30) / 2 = 66/2 = 33 cm.

Using Heron’s formula, we find the area S of this triangle:

S = √ (33 * (33 – 11) * (33 – 25) * (33 – 30)) = √ (33 * 22 * 8 * 3) = √ (3 * 11 * 2 * 11 * 8 * 3) = 11 * 3 * √ (2 * 8) = 33 * √16 = 33 * 4 = 132 cm ^ 2.

Using the formula for the area of a triangle through the radius r of the inscribed circle, we find r:

r = S / p = 132/33 = 4 cm.

Using the formula for the area of a triangle through the radius R of the circumscribed circle, we find R:

R = 11 * 25 * 30 / (4 * 132) = 8250/528 = 15.625 cm.

Answer: the area of the triangle is 132 cm ^ 2, the radius of the inscribed circle is 4 cm, the radius of the circumscribed circle is 15.625 cm.

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