The leg and hypotenuse of a right triangle are 12 and 15 cm, the distance from this point

The leg and hypotenuse of a right triangle are 12 and 15 cm, the distance from this point to the sides of the triangle is 5 cm. Find the distance from this point to the plane of the triangle

By the Pythagorean theorem, we determine the length of the BC leg.

BC ^ 2 = AB ^ 2 – AC ^ 2 = 225 – 144 = 81.

BC = 9 cm.

Point D and the vertices of the triangle form a pyramid, in which the apothems of the side faces are equal, then point D is projected from the center of the circle inscribed in the triangle ABC.

Determine the radius of the inscribed circle.

OH = r = (AC + BC – AB) / 2 = (12 + 9 – 15) / 2 = 6/2 = 3 cm.

In a right-angled triangle DOН, according to the Pythagorean theorem, DO ^ 2 = DН ^ 2 – OH ^ 2 = 25 – 9 = 16.

DO = 4 cm.

Answer: From point D to the plane of the triangle 4 cm.