In a right-angled triangle, one of the acute angles is 30 degrees, and the hypotenuse is 12cm. Find the legs of this triangle.

A right-angled triangle is a triangle that has a right angle (equal to 90 °).

The side AB opposite the right angle is called the hypotenuse, and the other two are called the legs.

To calculate the BC leg, we use the sine theorem. The sine of an acute angle in a right triangle is equal to the ratio of the opposite leg to the hypotenuse:

sin A = BC / AB;

BC = AB · sin A;

sin 30 ° = 1/2;

BC = 12 1/2 = 12/2 = 6 cm.

To calculate the length of the second leg, we apply the Pythagorean theorem:

AB ^ 2 = BC ^ 2 + AC ^ 2;

AC ^ 2 = AB ^ 2 – BC ^ 2;

AC ^ 2 = 12 ^ 2 – 6 ^ 2 = 144 – 36 = 108;

AC = √108 = 10.4 cm.

Answer: the length of the BC leg is 6 cm, the AC leg length is 10.4 cm.