Calculate the area of a triangle whose side lengths are 4 cm and 8 cm

Calculate the area of a triangle whose side lengths are 4 cm and 8 cm, and the angle formed by these sides is half the adjacent outer corner of the triangle.

1. A, B, C the vertices of the triangle. ∠1 – outer corner adjacent to ∠А. AB = 4 cm. BC = 8 cm.

S is the area of the triangle. AE is the height drawn to the BC side.

2. ∠1 = 2∠A by the problem statement. ∠1 = ∠В + ∠С (according to the theorem on the outer angle of a triangle).

3. ∠А + ∠В + ∠С = 180 ° (according to the theorem on the sum of the angles of a triangle).

4. Replace (∠В + ∠С) with 2∠А:

∠А + 2∠А = 180 °.

∠А = 60 °.

5. AE: AB = sine ∠A = sine 60 ° = √3 / 2.

AE = AB x √3 / 2 = 4 x √3 / 2 = 2√3 cm.

6. S = BC / 2 x AE = 8/2 x 2√3 = 8√3 cm²