Mathematics

Mathematics

Problem

Which of the following gives the area of a triangle whose side lengths are a, b, ca,\,b,\,c and whose semi-perimeter is s=a+b+c2s=\frac{a+b+c}{2}? a) abc4s\dfrac{abc}{4s} b) s(s−a)(s−b)(s−c)\sqrt{s(s-a)(s-b)(s-c)} c) 12absin⁡C\dfrac12 ab\sin C d) 12s(s−a)(s−b)(s−c)\dfrac12 \sqrt{s(s-a)(s-b)(s-c)}
Intuitive explanation

This is Heron's formula, which lets you find a triangle's area using only its three side lengths—no height or angles needed. The key insight is that the semi-perimeter ss naturally captures how "spread out" the sides are, and the product s(s−a)(s−b)(s−c)\sqrt{s(s-a)(s-b)(s-c)} elegantly encodes the geometry. Rather than measuring perpendicular distance, you're using how far each side falls short of the semi-perimeter to determine area.

Intuitive explanation

This is Heron's formula, which lets you find a triangle's area using only its three side lengths—no height or angles needed. The key insight is that the semi-perimeter ss naturally captures how "spread out" the sides are, and the product s(s−a)(s−b)(s−c)\sqrt{s(s-a)(s-b)(s-c)} elegantly encodes the geometry. Rather than measuring perpendicular distance, you're using how far each side falls short of the semi-perimeter to determine area.

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