Multiple choice

A lamp post standing at a point $A$ on a circular path of radius $r$ subtends an angle $\alpha$ at some point $B$ on the path, and $AB$ subtends an angle of $45^{0}$ at any other point on the path, the height of the lamp post is

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

In a circle, the angle subtended by a chord at the circumference is constant. If AB subtends 45 degrees, the radius r and chord AB are related by AB = 2r sin(45) = r*sqrt(2). The lamp post height h at A subtends alpha at B. h = AB * tan(alpha) = r*sqrt(2)*tan(alpha).

AI explanation

The chord AB subtends an angle of 45 degrees at a point on the alternate segment of the circle, so the central angle subtended by AB is 90 degrees. Using the properties of an isosceles right triangle, the distance AB equals the square root of 2 times the radius r. In the right triangle formed with the lamp post of height h at point A, the angle of elevation from point B is alpha, giving tan(alpha) equals h divided by AB. Substituting the value of AB gives tan(alpha) equals h divided by (the square root of 2 times r), so h equals the square root of 2 times r times tan(alpha).

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