The two sides of the triangle, the lengths of which are 3: 8, form an angle of 60 degrees.

The two sides of the triangle, the lengths of which are 3: 8, form an angle of 60 degrees. Find these sides if the third side is 21 cm.

Let us denote the length of one part of the side of the triangle by variable x. Then the lengths of the sides of the triangle are 3x and 8x, respectively.

According to the cosine theorem:

a ^ 2 = b ^ 2 + c ^ 2 – 2bc * cosα, where a is the length of the side opposite to the angle α (a = 21 cm), b is the length of the second side of the triangle (b = 3x), c is the length of the third side of the triangle (c = 8x), α = 60º.

21 ^ 2 = (3x) ^ 2 + (8x) ^ 2 – 2 * 3x * 8x * cos60º.

441 = 9x ^ 2 + 64x ^ 2 – 24x ^ 2.

441 = 49x ^ 2.

x ^ 2 = 441/49.

x = √9 = 3.

b = 3x = 3 * 3 = 9 cm.

c = 8x = 8 * 3 = 24 cm.