In a right-angled triangle, the hypotenuse is 10 and one of the acute angles is 30 degrees. Find the area of the triangle.

1) Find the first leg of a right-angled triangle. We know that in a triangle one of the acute angles is 30 °. In a right-angled triangle, the leg opposite to an angle of 30 ° is equal to half the hypotenuse.

1/2 * 10 = 10/2 = 5.

2). Find the second leg of the right-angled triangle. We apply the Pythagorean theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

10 ^ 2 = 5 ^ 2 + x ^ 2;

x ^ 2 = 10 ^ 2 – 5 ^ 2 = 100 – 25 = 75;

x = √75 = √ (25 * 3) = 5√3.

3) Find the area of ​​the triangle. The area of ​​a right-angled triangle is half the product of its legs.

S = 1/2 * 5 * 5√3 = (25√3) / 2 = 12.5 √3.

Answer. 12.5 √3.