In a right-angled triangle ABC, the angle C is 90 degrees; the smallest side is 2 times

In a right-angled triangle ABC, the angle C is 90 degrees; the smallest side is 2 times shorter than the largest, find the degree measure of one of the acute angles of the triangle.

Let x denote the length of the smallest side of a given right-angled triangle.

According to the condition of the problem, the smallest side of a given right-angled triangle is 2 times shorter than the largest, therefore, the length of the largest side of this right-angled triangle should be equal to 2x.

Since the largest side of any right-angled triangle is its hypotenuse, and the smallest side is the smaller of the legs, then in this triangle the length of the hypotenuse is 2x, and one of the legs is x.

Let α denote the angle opposite to the leg x.

Applying the theorem of sines, we get the relation:

2x / sin (90 °) = x / sin (α),

whence follows:

sin (α) = x * sin (90 °) / (2x) = x * 1 / (2x) = 1/2;

α = 30 °.

Answer: 30 °.