The height drawn from the vertex of the right angle C of a right-angled triangle divides the hypotenuse

The height drawn from the vertex of the right angle C of a right-angled triangle divides the hypotenuse into two parts: 4 and 9. Calculate the height.

Let us prove that triangle ACH is similar to triangle BCH.

Let the angle BAC = X0, then in the right-angled triangle ACH the angle ACH = (90 – X) 0.

Angle ACB = 90, then the angle BCH of the right-angled triangle BCH is equal to:

Angle BCH = (90 – ACH) = (90 – (90 – X)) – X0.

Then the triangles ACH and BCH are similar in acute angle.

Then in similar triangles ACH and BCH:

AH / CH = CH / BH.

CH ^ 2 = AH * BH = 9 * 4 = 36.

CH = 6 cm.

Answer: The length of the CH height is 6 cm.