In triangle ABC, AB = BC, angle CAB = 30 degrees, AE is the bisector, BE = 8 cm. Find the area of triangle ABC.

From the condition it is known that in a triangle ABC, the sides are equal to each other AB = BC, it is also known that the angle CAB = 30 °, AE is the bisector of the triangle, and BE = 8 cm.

Let’s calculate the area of triangle ABC.

We introduce the designations AB = BC = a and AC = b, then CE = a – 8.

Let’s write the theorem of sines:

a / sin 30 ° = b / sin 120 ° whence we express

b = a * sin 120 ° / sin 30 ° = a√3;

We apply the theorem on the bisector of an angle and compose the proportion:

a / b = 8 / (a – 8) or a / a√3 = 8 / (a – 8).

Find a:

a = 8 (1 + √3).

We are looking for an area:

S of triangle ABC = 0.5 a ^ 2 * sin 120 ° = 0.5 * 64 (1 + √3) ^ 2 * (√3 / 2) = 16√3 (1 + √3) ^ 2 = 32√ 3 (2 + √3) cm ^ 2.