Multiple choice

A pole of height $h$ stands at one corner of a park in the shape of an equilateral triangle. If $\alpha$ is the angle which the pole subtends at the midpoint of the opposite side, the length of each side of the park is:

  1. $\displaystyle \frac{\sqrt{3}}{2}h\cot\alpha$
  2. $\displaystyle \frac{2}{\sqrt{3}}h\cot\alpha$
  3. $\displaystyle \frac{\sqrt{3}}{2}h\tan\alpha$
  4. $\displaystyle \frac{2}{\sqrt{3}}h\tan\alpha$
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B Correct answer
Explanation

Let the side of the equilateral triangle be 'a'. The distance from the midpoint of a side to the opposite vertex is (sqrt(3)/2) * a. Using trigonometry in the right triangle formed by the pole and this distance, tan(alpha) = h / ((sqrt(3)/2) * a). Solving for 'a' gives a = (2h / sqrt(3)) * cot(alpha).

AI explanation

Let the side length of the equilateral triangular park be s and the pole of height h be at one corner. The midpoint of the opposite side is at a perpendicular distance of s * sqrt(3) / 2 from the pole, because the altitude of the equilateral triangle is s * sqrt(3) / 2. The angle alpha subtended by the pole at this midpoint is found using tan alpha = h / (s * sqrt(3) / 2). Rearranging this equation to solve for s gives s = (2 / sqrt(3)) * h * cot alpha.

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