Find the area of a right-angled triangle if its acute angles are 1: 2 and the hypotenuse is 8cm.

1. Vertices of triangle A, B, C. AB = 8 cm – hypotenuse. ∠С = 90 °.

2. Suppose ∠А: ∠В = 1: 2, that is ∠В = 2∠А.

3. ∠А + ∠В + ∠С = 180 °.

∠А + 2∠А + 90 ° = 180 °.

3∠A = 90 °.

∠А = 30 °.

4. BC = AB / 2, since, according to the properties of a right-angled triangle, the leg located

opposite an angle of 30 ° is 1/2 hypotenuse.

BC = 8: 2 = 4 cm.

5. AC = √AB² – BC² = √8² – 4² = √64 – 16 = √48 = 4√3 cm.

6. The area of the triangle = BC x AC / 2 = 4 x 4√3 / 2 = 8√3 cm².

Answer: the area of the triangle is 8√3 cm².