The height CD, drawn from the vertex of the right angle C of the right-angled triangle ABC, is 12 cm

The height CD, drawn from the vertex 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?

The height of the СD divides the ABC triangle into two right-angled triangles, the ВСD and the AСD.
Let us prove that the triangles ВСD and AСD are similar.
Let the angle CAB = α0, then the angle ABC = (90 – α) 0.
In a right-angled VSD triangle, the ВСD angle = (90 – AB) = (90 – (90 – α)) = α0.
Then the angle BCA = CAB = СAD, and then the right-angled triangles of the ВСD and AСD are similar in acute angle.
From the similarity of triangles СD / AD = ВD / СD.
ВD = СD2 / AD = 144/9 = 16 cm.
Answer: The length of the ВD segment is 16 cm.