Three circles of radii 8 cm, 5 cm and 5 cm touch each other externally. The area of the triangle formed by the line segments joining the centers of the three circles is

To solve this problem, we need to use the formula for the area of a triangle when the lengths of all three sides are known. This is known as Heron's formula.

Step 1: Find the lengths of the sides of the triangle. The sides of the triangle are the distances between the centers of the circles. Since the circles touch each other externally, the distance between the centers of two circles is equal to the sum of their radii. Therefore, the sides of the triangle are 8 cm + 5 cm = 13 cm, 5 cm + 5 cm = 10 cm, and 8 cm + 5 cm = 13 cm.

Step 2: Calculate the semi-perimeter of the triangle. The semi-perimeter (s) of a triangle with sides a, b, and c is given by the formula s = (a + b + c) / 2. Substituting the lengths of the sides, we get s = (13 cm + 10 cm + 13 cm) / 2 = 18 cm.

Step 3: Use Heron's formula to find the area of the triangle. Heron's formula for the area (A) of a triangle is A = sqrt[s(s - a)(s - b)(s - c)]. Substituting the values we have, we get A = sqrt[18 cm(18 cm - 13 cm)(18 cm - 10 cm)(18 cm - 13 cm)] = sqrt[18 cm * 5 cm * 8 cm * 5 cm] = sqrt[36000 cm^2] = 60 cm^2.

Therefore, the area of the triangle formed by the line segments joining the centers of the three circles is 60 cm^2.