In triangle ABD with sides AB is equal to 16 AD is equal to 15 BD is equal to 9, the bisector DK is drawn to find AK and BK.

Since the bisector of the angle of a triangle divides the opposite side into segments proportional to the adjacent sides, then:

AB / BK = AD / AK.

Let the length of the segment AK = X cm, then the length of the segment BK will be equal to (15 – X) cm.

Then: AB / (16 – X) = AD / X.

9 / (16 – X) = 15 / X.

15 * (16 – X) = 9 * X.

240 – 15 * X = 9 * X.

24 * X = 240.

X = 240/24 = 10 cm.

AK = 10 cm.

BK = 16 – 10 = 6 cm.

Answer: The length of the segment AK is 10 cm, BK is 6 cm.