Point M is removed from each side of the isosceles trapezoid at a distance of 16 cm. The base of the trapezoid is 18

Point M is removed from each side of the isosceles trapezoid at a distance of 16 cm. The base of the trapezoid is 18 and 32 cm. Find the distance from point M to the plane of the trapezoid.

If a point does not lie in the plane of a convex polygon and its distance from all sides of the polygon is the same, then the projection of this point onto the plane is the center of the circle inscribed in the polygon.

A circle can be inscribed in a trapezoid if the sum of the sides of the trapezoid is equal to the sum of its bases.

AD + BC = AB + CD = 32 + 18 = 50 cm.

Since the trapezoid is isosceles, then AB = CD.

Then 2 * AB = 50.

AB = CD = 50/2 = 25 cm.

Let us draw from the vertex B of the trapezium the height BK, which cuts off the segment AK on the basis of AD, the length of which is equal to the half-difference of the base.

AK = (AD – BC) / 2 = (32 – 18) / 2 = 14/2 = 7 cm.

From the rectangular triangle ABK, according to the Pythagorean theorem, we determine the height BK.

BK ^ 2 = AB ^ 2 – AK ^ 2 = 25 ^ 2 – 7 ^ 2 = 625 – 49 = 576.

BK = 24 cm.

The height of the trapezoid is equal to the diameter of the inscribed circle, then R = BK / 2 = 24/2 = 12 cm.

Consider a right-angled triangle MOH, in which, by condition, the hypotenuse MH = 16 cm, and the leg OH is equal to the radius of the circle, OH = 12 cm.

MO ^ 2 = MH ^ 2 – OH ^ 2 = 16 ^ 2 – 12 ^ 2 = 256 – 144 = 112.

MO = 4 * √7 cm.

Answer: The distance from point M to the plane is 4 * √7 cm.