Question

Find the area of an equilateral triangle PQRwhose each side is units

Answer 1

$\text{Let the side of the equilateral triangle = a} \\ \: \: \: \: \: \: \: s = \frac{1}{2} (a + a + a) = \frac{3}{2} a \\ \text{Area} = \sqrt{s(s - a)(s - b)(s - c)} \\ \: \: \: \: \: \: \: \: \: \: = \sqrt{ \frac{3}{2} a( \frac{3}{2} a - a)( \frac{3}{2} a - a)( \frac{3}{2} a - a)} \\ \: \: \: \: \: \: \: \: \: \: = \sqrt{ \frac{3}{2}a( \frac{a}{2} )( \frac{a}{2} )( \frac{a}{2} )} \\ \: \: \: \: \: \: \: \: \: \: = \sqrt{ \frac{3 \times {a}^{2} \times {{a}^{2}}}{ {2}^{2} \times {2}^{2} } } = \frac{a \times a}{2 \times 2} \sqrt{3} \\ \: \: \: \: \: \: \: \: \: \: = \frac{ {a}^{2} }{4} \sqrt{3} = \frac{ \sqrt{3} }{4} {a }^{2} \\ \text{Thus, area of an equilateral triangle} = \frac{ \sqrt{3} }{4} \text{side} {}^{2}$
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