The bisector BK is drawn in the triangle ABC. Determine the length of the segment AK

The bisector BK is drawn in the triangle ABC. Determine the length of the segment AK, if it is known that AB = 7.5, BC = 6, CK = 4.

We will use the well-known property of the bisector of any triangle, namely, the fact that the ratio of the lengths of the segments AK and KС, into which the bisector of this triangle divides the segment AC is equal to the ratio of the length of the side AB to the length of the side BC of this triangle:

| AK | / | KC | = | AB | / | BC |.

Since in this triangle | AB | = 7.5, | BC | = 6 and | KC | = 4, we can make the following equation:

| AK | / 4 = 7.5 / 6,

solving which, we get:

| AK | = 4 * 7.5 / 6 = 2 * 7.5 / 3 = 2 * 2.5 = 5.

Answer: the length of the AK segment is 5.