In rectangle ABC, angle C = 90 degrees, AC = 6cm AB = 9cm CD height. Find BD

By the Pythagorean theorem, we determine the length of the BC leg.

BC ^ 2 = AB ^ 2 – AC ^ 2 = 81 – 36 = 45.

BC = 3 * √5 cm.

Let the length of the segment BD = X cm, then AD = (9 – X) cm.

In right-angled triangles ACD and BCD, we express the leg CD by the Pythagorean theorem and equate the obtained equalities.

CD ^ 2 = AC ^ 2 – AD ^ 2 = 36 – (9 – X) ^ 2 = 36 – 81 + 18 * X – X ^ 2 = -X ^ 2 + 18 * X – 45.

CD ^ 2 = BC ^ 2 – BD ^ 2 = 45 – X ^ 2.

Then: -X2 + 18 * X – 45 = 45 – X ^ 2.

18 * X = 90.

X = BD = 90/18 = 5 cm.

Answer: The length of the VD segment is 5 cm.