Triangle ABC has side AB 15 cm less than BC. Find the perimeter of the triangle if the bisector

Triangle ABC has side AB 15 cm less than BC. Find the perimeter of the triangle if the bisector of angle B divides the AC side into 4 cm and 14 cm segments.

Let the length of the side AB of the triangle ABC be equal to X cm, then, according to the condition, the length of the side BC is equal to (X + 15) cm.

Since BH is the bisector of angle B, by its property, it divides the base of the AC into segments proportional to the adjacent sides.

Then:

AH / AB = CH / BC.

4 / X = 14 / (X +15).

14 * X = 4 * (X + 15).

14 * X – 4 * X = 60.

10 * X = 60.

X = AB = 60/10 = 6 cm.

BC = 6 + 15 = 21 cm.

Determine the perimeter of the triangle.

P = 6 + 21 + (4 + 14) = 45 cm.

Answer: The perimeter of the triangle is 45 cm.