Multiple choice

The angles of elevation of a church from two points at distance $\displaystyle d_{1}$ and $\displaystyle d_{2}$ from the floor of the church on the side are complementary The height of the church is

  1. $\displaystyle \sqrt{d_{1}d_{2}}$
  2. $\displaystyle \sqrt{d_{1}/d_{2}}$
  3. $\displaystyle d_{1}d_{2}$
  4. $\displaystyle d_{1}/d_{2}$
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A Correct answer
Explanation

If the height is h, the two tangent values are h/d1 and h/d2. Complementary angles have tangent values whose product is 1, so h^2/(d1d2) = 1 and h = sqrt(d1d2).

AI explanation

Let the angles of elevation be theta and 90 degrees minus theta, since they are complementary. We have tan(theta) = height / d1 and tan(90 degrees - theta) = height / d2, which means cot(theta) = height / d2. Multiplying these two equations gives tan(theta) * cot(theta) = (height / d1) * (height / d2). Since tan(theta) * cot(theta) = 1, we get height^2 = d1 * d2, and taking the square root yields the result sqrt(d1 * d2).

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