10.2 Heron's Formula: The Rigorous Derivation

Identify sides: $a=8, b=11, c=13$.

Calculate semi-perimeter $s$: $s = \frac{8+11+13}{2} = \frac{32}{2} = 16$ cm.

Calculate terms $(s-a), (s-b), (s-c)$: $s-a = 16-8 = 8$ $s-b = 16-11 = 5$ $s-c = 16-13 = 3$

Apply Heron's Formula: $A = \sqrt{16 \times 8 \times 5 \times 3}$

Simplify square root: $A = \sqrt{16 \times 4 \times 2 \times 5 \times 3} = 4 \times 2 \sqrt{2 \times 5 \times 3} = 8\sqrt{30}$.

Calculate decimal (optional): $8 \times 5.477 \approx 43.82 \text{ cm}^2$.