From point D, lying on the hypotenuse AB of the right-angled triangle ABC

From point D, lying on the hypotenuse AB of the right-angled triangle ABC, the perpendicular DE is dropped to BC. Find AC if BC = 12cm, BE = 8cm, DE = 6cm.

Let us prove that triangles ABC and DBE are similar.

Angles ABC and DEB of triangles ABC and DBE are equal to 90 by condition, and angle B is common in construction, then triangle ABC is similar to triangle DBE by the first sign of similarity of right-angled triangles – by an acute angle.

Then with respect to the similar sides of similar triangles:

AC / DE = BC / BE.

AC / 6 = 12/8.

AC = 12 * 6/8 = 9 cm.

Answer: Speaker length = 9 cm.