What is PN equal to?
Consider the following for the two (02) items that follow : The top (M) of a tower is observed from three points P, Q and R lying in a horizontal straight line which passes directly along the foot (N) of the tower. The angles of elevations of M from P, Q and R are 30^{\circ}, 45^{\circ} and 60^{\circ} respectively. Let PQ=a and QR=b.
- A. (\frac{3-\sqrt{3}}{2})a
- B. (\frac{3+\sqrt{3}}{2})a ✓
- C. (\frac{3-\sqrt{3}}{4})a
- D. (\frac{3+\sqrt{3}}{4})a
Correct Answer: B. (\frac{3+\sqrt{3}}{2})a
Explanation
Let MN = h. From the right triangles MNR, MNQ, MNP, we have RN = h\cot 60^{\circ} = h/\sqrt{3}, QN = h\cot 45^{\circ} = h, and PN = h\cot 30^{\circ} = h\sqrt{3}. We know PQ = PN - QN = h(\sqrt{3}-1) = a \implies h = \frac{a}{\sqrt{3}-1} = \frac{a(\sqrt{3}+1)}{2}. So, PN = h\sqrt{3} = a(\frac{3+\sqrt{3}}{2}).