The height BD of a right-angled triangle ABC is 24 cm and cuts off a segment DC equal to 18 cm

The height BD of a right-angled triangle ABC is 24 cm and cuts off a segment DC equal to 18 cm from the hypotenuse Find AB and cos A.

According to metric ratios in a right-angled triangle, the square of the height is equal to the product of the projections of the legs and the hypotenuse. DC and AC are projections, therefore:
BD ^ 2 = AD * DC;
AD = BD ^ 2 / DC.
AD = 24 ^ 2/18 = 32 (cm)
AC = 31 + 18 = 50 (cm)
According to the same metric ratios, the square of the leg is equal to the product of the hypotenuse and the projection of this leg to the hypotenuse:
AB ^ 2 = AD * AC
AB ^ 2 = 32 * 50 = 1600
AB = 40 cm.
The cosine of the angle is the ratio of the adjacent leg to the hypotenuse:
cos A = AB / AC
cos A = 40/50 = 0.8.