Find the area of a triangle whose three sides measure 13 cm, 14 cm, and 15 cm. Use Heron's formula and give the final area in square centimetres (cm^2).

Introduction / Context:
This problem tests Heron's formula, which is used to find the area of a triangle when all three sides are known. It avoids needing the height explicitly. The method is especially useful for scalene triangles (all sides different). The key steps are finding the semi-perimeter and then substituting into the formula.

Given Data / Assumptions:

• Sides: a = 13 cm, b = 14 cm, c = 15 cm
• Semi-perimeter s = (a + b + c)/2
• Heron's formula: Area = sqrt(s(s-a)(s-b)(s-c))

Concept / Approach:
Compute s first. Then compute s-a, s-b, s-c. Multiply all four values and take the square root. This gives the area in cm^2.

Step-by-Step Solution:
s = (13 + 14 + 15)/2 = 42/2 = 21 s - a = 21 - 13 = 8 s - b = 21 - 14 = 7 s - c = 21 - 15 = 6 Area = sqrt(21*8*7*6) 21*8 = 168, 7*6 = 42, so product = 168*42 = 7056 Area = sqrt(7056) = 84 cm^2

Verification / Alternative check:
The computed value is exact because 7056 is a perfect square (84^2). So no rounding is needed, and the area 84 cm^2 is precise.

Why Other Options Are Wrong:
64 cm^2 and 44 cm^2: result from incorrect semi-perimeter or incorrect multiplication in Heron's formula. 22 cm^2: far too small for a triangle with sides around 14 cm. 91 cm^2: close but incorrect; likely from arithmetic error before taking the square root.

Common Pitfalls:
Forgetting to divide perimeter by 2 to get semi-perimeter. Using (a+b+c) instead of s in the formula. Making multiplication errors before taking the square root.

Final Answer:
Area = 84 cm^2