Question

The topmost point of a perfectly vertical pole is marked A. The pole stands on a flat ground at point D. The points B and C are somewhere between A and D on the pole. From a point E, located on the ground at a certain distance from D, the points A, B and C are at angles of 60, 45 and 30 degrees respectively. What is AB : BC : CD?

Accepted Answer

D. Let $ED = x\sqrt{3}$. In $	riangle CDE$: $	an 30^\circ = \frac{CD}{ED} \implies CD = x\sqrt{3} 	an 30^\circ = x$. In $	riangle BDE$: $	an 45^\circ = \frac{BD}{ED} \implies BD = x\sqrt{3}$. In $	riangle ADE$: $	an 60^\circ = \frac{AD}{ED} \implies AD = 3x$. $BC = BD - CD = x\sqrt{3} - x = x(\sqrt{3}-1)$. $AB = AD - BD = 3x - x\sqrt{3} = x(3-\sqrt{3})$. Ratio $AB : BC : CD = (3-\sqrt{3}) : (\sqrt{3}-1) : 1$.

Ratio of Heights on Vertical Pole

Explanation

Practise this under exam conditions