In a right-angled triangle BCD from point M, lying on the hypotenuse BC, the perpendicular MN

In a right-angled triangle BCD from point M, lying on the hypotenuse BC, the perpendicular MN is dropped to the leg BD. Find the hypotenuse if MN = 12 cm, CD = 18 cm, MC = 8 cm.

Let us prove that triangles CBD and MBN are similar.

Triangle CBD is rectangular by condition, and MBN by construction, since MN is perpendicular to BD. Angle B in triangles is common, then right-angled triangles are similar in acute angle.

Let the length of the segment BM = X cm, then the length of the hypotenuse BC = (X + 8) cm.

Then: CB / CD = BM / MN.

(X + 8) / 18 = X / 12.

18 * X = 12 * X + 96.

6 * X = 96.

X = BM = 96/6 = 16 cm.

Then BC = 16 + 8 = 24 cm.

Answer: The length of the hypotenuse is 24 cm.