In a right-angled triangle KPF, the height KD of triangle KPF is 24 cm and cuts off a segment

In a right-angled triangle KPF, the height KD of triangle KPF is 24 cm and cuts off a segment DF equal to 18 cm from the hypotenuse PF, find KP and cos of the angle P

Since the height KD is drawn to the hypotenuse from a right angle, then:

KD ^ 2 = PD * DF, then:

PD = KD ^ 2 / DF = 576/18 = 32 cm.

Then the length of the hypotenuse is PF = PD + DF = 32 + 18 = 50 cm.

From the right-angled triangle KPD we determine the length of the hypotenuse KP.

KP ^ 2 = PD ^ 2 + KD ^ 2 = 32 ^ 2 + 24 ^ 2 = 1024 + 576 = 1600.

KP = 40 cm.

Determine the cosine of the angle KPF.

CosKPF = KP / PF = 40/50 = 4/5.

Answer: The length of the side of the KP is 40 cm, CosP = 4/5.