In an isosceles triangle ABC, the base of a is 16 cm. The angle C is 120 degrees. find the bisector of the angle ABC.

1. From the vertex C of the triangle ABC we drop the bisector CH. CH will also be the median and height, since in an isosceles triangle the bisector drawn to the base is both the median and the height.

2. Consider a triangle НСВ – rectangular (СН – height, angle Н = 90 degrees)

1) The angle of the НСВ will be = 60 degrees (CH is the bisector)
2) Then the angle СВН = 30 degrees. And the leg, which lies opposite an angle of 30 degrees (in our case, CH, which we just need to find) is equal to half of the hypotenuse, that is, CB. But its meaning is still unknown to us.
3) To find the СВ, use the sine theorem:

sinС       sinA
—— = ——–
AB         СВ

Angle A is calculated from the sum of the angles of the triangle: (180 – 120) / 2 = 30 degrees. (angles in an isosceles triangle at the base are equal)

We express CB: AB * sinA / sinC = 16 * 1/2 / √3 / 2 =
CH = 1/2 CB = 2√3
Answer: bisector of angle ACB = 4√3 cm