In a right triangle, the hypotenuse is 2 and one of the acute angles is 45∘. Find the area of the triangle.

It is known:

Right-angled triangle ABC;
Hypotenuse AB = 2;
One angle = 45 °.
Find the area of the triangle.

If one angle in a right triangle is 45 °, then the second angle is also 45 °.

Hence, angle A = angle B = 45 °.

If the angles are equal, then AC = BC.

Find one leg of a right-angled triangle if the hypotenuse and the angles of the triangle are known.

sin a = BC / AB;

BC = AB * sin a = AB * sin 45 ° = 2 * √2 / 2 = √2;

Hence, AC = BC = √2;

The area of a right-angled triangle is half the product of the legs.

S = (AC * BC) / 2 = √2 * √2 / 2 = √4 / 2 = 2/2 = 1;

Answer: S = 1.