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Question :

  • In an equilateral triangle with SIDE a , PROVE that area = √3a²/4

To Prove :

  • Area of equilateral triangle = √3a²/4

Proof :

FIRST CHECK attachment

By using Pythagoras theorem

\tt \implies{a}^{2} = {h}^{2} + { \bigg(\frac{a}{2} \bigg)}^{2} \\ \\\implies \tt {a}^{2} = {h}^{2} + \frac{ {a}^{2} }{4} \\ \\ \tt \implies {h}^{2} = {a}^{2} - \frac{ {a}^{2} }{4} \\ \\ \implies \tt {h}^{2} = \frac{ {4a}^{2 } - {a}^{2} }{4} \\ \\\implies \tt {h}^{2} = \frac{3 {a}^{2} }{4} \\ \\\implies \tt h = \sqrt { \frac{3 {a}^{2} }{4} } \\ \\ \implies\tt h = \frac{ \sqrt{3} a}{2}

Now

\large\boxed{ \tt Area_{triangle} = \frac{1}{2} \times b \times h} \\ \\ \tt \implies \frac{1}{2} \times a \times \frac{ \sqrt{3} a}{2} \\ \\ \tt \implies \frac{ \sqrt{3} {a}^{2} }{4}

Hence Proved


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