The height CD, drawn from the top of the right angle C of the right-angled triangle ABC

The height CD, drawn from the top of the right angle C of the right-angled triangle ABC, is 12 cm, and the segment AD = 9 cm. What is the segment BD?

Let us prove that triangles ACD and BCD are similar.

Let the angle CAB = α, then the angle ACD = 90 – α.

Angle ВСD = АВС – АСD = 90 – (90 – α) = α.

Angle CAD = BCD, right-angled triangles ACD and BCD are similar in acute angle.

Then AD / CD = CD / BD.

CD ^ 2 = AD * BD.

ВD = СD ^ 2 / АD = 12 ^ 2/9 = 144/9 = 16 cm.

Answer: The length of the segment BD = 16 cm.