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Question Which has a greater area — an equilateral triangle or a square of the same side length as the triangle? Which has a greater area: two identical equilateral triangles together or a square of the same side length as the triangle? Give reasons.
Area of equilateral triangle = \(\frac{\sqrt{3}}{4} a^2\) Area of square = a 2 ⇒ \(\frac{\sqrt{3}}{4} a^2\) < a 2 So, the area of a square is greater than the area of an equilateral triangle of the same side length. Area of two identical equilateral triangles = \(\frac{\sqrt{3}}{4} a^2+\frac{\sqrt{3}}{4} a^2\) = \(\frac{2 \sqrt{3}}{4} a^2\) = \(\frac{\sqrt{3}}{2} a^2\) Area of square of side length a = a 2 Clearly, \(\frac{\sqrt{3}}{2} a^2\) < a 2 So, the area of a square is greater than the area of two identical equilateral triangles. Rhombus & Trapezium Figure It Out (Pages 169-170)