The sides of the triangle are 9 cm, 10 and 17 cm. Find the distance from the plane of the triangle

The sides of the triangle are 9 cm, 10 and 17 cm. Find the distance from the plane of the triangle to the center of the ball, which touches all sides of the triangle, if the radius of the ball is 3 cm.

Let us determine the radius О1Н of the inscribed circle formed by the tangency of the sphere of the sides of the triangle ABC.

The semi-perimeter of the triangle ABC is equal to: p = (AC + BC + AB) / 2 = (17 + 10 + 9) / 2 = 18 cm.

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

Sас = √18 * (18 – 17) * (18 – 10) * (18 – 9) = √1296 = 36 cm2.

Let’s define the radius О1Н.

О1Н = Savs / p = 36/18 = 2 cm.

Triangle O1OH is rectangular, then by the Pythagorean theorem, OO1 ^ 2 = OH ^ 2 – OO1 ^ 2 = 9 – 4 = 5.

OO1 = √5 cm.

Answer: From the center of the ball to the plane of the triangle √5 cm.