Question

prove that area of equilateral triangle is root 3 by 4 multiplied by xsquare where x is aside of triangle

Answer 1

$s = \frac{x + x + x}{2} = \frac{3x}{2} ....since \: all \: sides \: are \: equal \\ \\ area = \sqrt{s(s - a)(s - b)(s - c)} ...herons \: formula \\ area = \sqrt{ \frac{3x}{2}( \frac{3x}{2} - x)( \frac{3x}{2} - x)( \frac{3x}{2} - x) } \\ area = \sqrt{ \frac{3x}{2} \times \frac{3x - 2x}{2} \times \frac{3x - 2x}{2} \times \frac{3x - 2x)}{2} } \\ area = \sqrt{ \frac{3x}{2} \times \frac{x}{2} \times \frac{x}{2} \times \frac{x}{2} } \\ area = \frac{x}{2} \sqrt{ \frac{ {3x}^{2} }{4} } \\ area = \frac{ \sqrt{3} {x}^{2} }{4}$