In triangle ABC, side AB is 8 cm, side BC is 2 cm, angle ABC is 30 degrees

In triangle ABC, side AB is 8 cm, side BC is 2 cm, angle ABC is 30 degrees. BD is the bisector of the angle ABC. Find the area of a triangle ABD.

Determine the area of the triangle ABC.

Savs = AB * BC * Sin30 / 2 = 8 * 2 * (1/2) / 2 = 4 cm2.

Since BD is the bisector of angle B, then the angle ABD = CBD = 30/2 = 150.

The area of the triangle ABD is equal to:

Savd = AB * BD * Sin15 / 2.

The area of the triangle CВD is equal to:

Ssvd = CВ * ВD * Sin15 / 2.

Then Svd / Ssvd = AВ / СВ = 8/2 = 4.

Svd = 4 * Ssvd. (1)

Savs = Savd + Ssvd = 4 cm2. (2)

Let’s solve the system of equations 1 and 2.

4 * Ssvd + Ssvd = 4.

Ssvd = 4/5 = 0.8 cm2.

Savd = Savs – Ssvd = 4 – 0.8 = 3.2 cm2.

Answer: The area of triangle ABD is 3.2 cm2.