The height CD of the right-angled triangle ABC cuts off the hypotenuse AB = 9 cm, the segment AD = 4 cm

The height CD of the right-angled triangle ABC cuts off the hypotenuse AB = 9 cm, the segment AD = 4 cm. Prove that the triangle ABC is similar to the triangle ACD. Find AC.

Since CD is height, triangles ABC and ACD are rectangular.

Let the angle ABC = X0, then the angle BCD = (90 – X) 0.

Then the angle ACD = 90 – (90 – X) = X0.

The angle ABC is equal to the angle ACD, then the right-angled triangles ABC and ACD are similar in acute angle, which was required to be proved.

Then: AB / AC = AC / AD.

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

AC = 6 cm.

Answer: The length of the speaker is 6 cm.