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Heron's Formula

If the sides ...

Question

If the sides of triangle $A B C$ satisfy the relation $a + b - c = 2$ and $2 a b - c^{2} = 4$ , then square of the area of triangle is

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Solution

Given:
$a + b - c = 2$ and $2 a b - c^{2} = 4$
$\Rightarrow 2 a b - 4 = (a + b - 2)^{2} = c^{2}$
$\Rightarrow 2 a b - 4 = a^{2} + b^{2} + 4 + 2 a b - 4 a - 4 b$
$\Rightarrow a^{2} + b^{2} - 4 a - 4 b + 8 = 0$
$\Rightarrow (a - 2)^{2} + (b - 2)^{2} = 0$
$∴ a = b = 2$
$\Rightarrow c = 2$
Equilateral triangle Area $Δ = \frac{\sqrt{3}}{4} \times 2^{2} = \sqrt{3}$
Square of the area $= Δ^{2} = 3$

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Heron's Formula

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