Question 310745
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The apothem of an equilateral triangle forms a 30-60-90 right triangle where the apothem is the long leg, one side of the triangle is the hypotenuse, and one-half of the side is the short leg.


With a tip o'the hat to Mr. Pythagoras, letting *[tex \Large x] represent the measure of the hypotenuse, we can say:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x^2\ =\ \left(\frac{x}{2}\right)^2\ +\ 16]


And a little algebra and rationalizing the denominator gets us to:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \frac{8\sqrt{3}}{3}]


Hence the measure of the short side of the triangle is:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{x}{2}\ =\ \frac{4\sqrt{3}}{3}]


The 30-60-90 triangle is one-half of the equilateral triangle, so:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ A\ =\ 2\left(\frac{4\,\cdot\,\frac{4\sqrt{3}}{3}}{2}\right)\ =\ \frac{4\,\cdot\,4\sqrt{3}}{3}\ =\ \frac{16\sqrt{3}}{3}\ \approx\ 9.2] feet.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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