The legs of a rectangular triangle ABC are 12cm and 16cm. From the vertex of the right angle C

The legs of a rectangular triangle ABC are 12cm and 16cm. From the vertex of the right angle C, a perpendicular CM, equal to 28cm, is drawn to the plane of the triangle. Find the distance from point M to the hepotenuse.

1. Draw the CH height to the hypotenuse. Line MC, perpendicular to plane ABC, is perpendicular to line AB, lying in this plane.

2. Line AB is perpendicular to line MC and the height of the triangle CH, so it is perpendicular to the plane MCH and the line MH lying in this plane. Therefore, MH is the perpendicular dropped from the point M to the hypotenuse:

AB = √ (AC ^ 2 + BC ^ 2);
AB = √ (16 ^ 2 + 12 ^ 2) = √ ((4 * 4) ^ 2 + (4 * 3 ^ 2) = 4√ (4 ^ 2 + 3 ^ 2) = 4√25 = 20;
AB * CH = AC * BC;
CH = AC * BC / AB = 16 * 12/20 = 9.6;
MH = √ (MC ^ 2 + HC ^ 2);
MH = √ (28 ^ 2 + 9.6 ^ 2) = 4√ (7 ^ 2 + 2.4 ^ 2) = 4√ (49 + 5.76) = 4√54.76 = 4 * 7.4 = 29.6.
Answer: 29.6 cm.