Triangle Area from Three Sides — Heron’s Formula: Find the area of a triangle whose sides are 8 cm, 10 cm, and 12 cm. Express the answer in simplified radical form.

Introduction / Context:
Heron’s formula computes the area of any triangle from its three side lengths, avoiding angle or height calculations. The key steps are finding the semiperimeter and then evaluating the square root expression carefully to keep radicals simplified.

Given Data / Assumptions:

• Sides: a = 8 cm, b = 10 cm, c = 12 cm
• All sides satisfy triangle inequality (valid triangle)
• Semiperimeter s = (a + b + c)/2

Concept / Approach:
Heron’s formula: Area = √[s(s − a)(s − b)(s − c)]. Compute s, the three differences, and multiply; then take the square root. Simplify the radical by extracting perfect squares to match common answer forms.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 8√63 and 7√93 do not equal 15√7.
• 6√53 is unrelated to the factorization of 1575.
• 15√7 is equivalent but not listed as the primary option; 5√63 is the same value in the provided form.

Common Pitfalls:

• Losing factors when simplifying √1575; factor completely to extract 9 and 25.

Final Answer:
5√63 sq cm (which equals 15√7 sq cm).