prove that area of an equilateral triangle is root 3/4 a^{2 , where a is thee side of the triangle}

A line perpendicular to the base through the opposite vertex bisects the base. This forms two right triangles. Each right triangle has a leg of length x/2, another of length h, and the hypotenuse of length x. The Pythagorean Theorem tells us x^2 = (x/2)^2 + h^2. x² = (x/2)² + h² x² - (x/2)² = h² x² - x² / 4 = h² 4x² - x² = 4h² 3x² / 4 = h² (√3 / 2) x = h The area of each right triangle is (1 / 2) * (x / 2) * h. The area of the equilateral triangle is 2 times that. A = 2 * (1 / 2) * (x / 2) * h A = (x / 2) * h A = (x / 2) * (√3 / 2) x A = (√3 / 4) x²

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