The height CD of the right-angled triangle ABC cuts off the segment AD equal to 4 cm from the hypotenuse

The height CD of the right-angled triangle ABC cuts off the segment AD equal to 4 cm from the hypotenuse AB equal to 9 cm. Prove that triangle ABC is similar to ACD and find AC.

Let the value of the angle ABC of the triangle ABC = X0, then the value of the angle BAC = (90 – X) 0.

In a right-angled triangle AСD, the angle of AСD = (180 – ADС – СAD) = (180 – 90 – (90 – X)) = X0.

Then the angle AСD = ABC, and therefore the right-angled triangles ABC and AСD are similar in acute angle, which was required to be proved.

Then in similar triangles ABC and AСD:

AB / AC = AC / AD.

AC^2 = AB * AD = 9 * 4 = 36.

AC = 6 cm.

Answer: The length of the AC segment is 6 cm.