One side of the triangle is 6 cm smaller than the other, and the angle between them is 60 °.

One side of the triangle is 6 cm smaller than the other, and the angle between them is 60 °. Find the perimeter if its third side is 14 cm.

Find the lengths of the first and second sides of this triangle.

Let x denote the length of the first side of this triangle.

According to the condition of the problem, the first side of this triangle is 6 cm less than its second side, therefore, the length of the second side of this triangle is x + 6.

According to the condition of the problem, the angle between the first and second sides is 60 °, and the third side of this triangle is 14 cm.

Using the cosine theorem, we get the following equation:

x ^ 2 + (x + 6) ^ 2 – 2x * (x + 6) * cos (60 °) = 14 ^ 2.

We solve the resulting equation:

x ^ 2 + (x + 6) ^ 2 – 2x * (x + 6) * (1/2) = 196.

x ^ 2 + x ^ 2 + 12x + 36 – x ^ 2 – 6x = 196;

x ^ 2 + 6x + 36 – 196 = 0;

x ^ 2 + 6x – 160 = 0;

x = -3 ± √ (9 + 160) = -3 ± √169 = -3 ± 13;

x1 = -3 – 13 = -16;

x2 = -3 + 13 = 10.

Since the length of the side of the triangle is positive, the value x = -16 is not suitable.

Thus, the first side of the triangle is 10 cm.

Find the second side:

x + 6 = 10 + 6 = 16 cm.

Find the perimeter of the triangle:

10 + 16 + 14 = 10 + 30 = 40 cm.

Answer: the perimeter of the triangle is 40 cm.