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?

• A. \((3+ \sqrt{3}): (1+ \sqrt{3}):1\)
• B. \((3- \sqrt{3}):1:(\sqrt{3}-1)\)
• C. \(1:1:1\)
• D. \((3- \sqrt{3}):(\sqrt{3}-1):1\)
• E. \((\sqrt{3}-1):1:(3- \sqrt{3})\)

Answer 1

The Correct Option is D

The topmost point of a vertical pole is A, bottom is D. Points B and C are between A and D. From a point E on the ground: 
∠ of elevation to A = 60°, 
∠ of elevation to B = 45°, 
∠ of elevation to C = 30°.  

Find the ratio \( AB : BC : CD \).

Step 1: Setup

Let ED = \(x\), and AD = \(h\). 
\(\tan 60^\circ = \frac{h}{x} \Rightarrow h = x\sqrt{3}\) 
\(\tan 45^\circ = \frac{BD}{x} \Rightarrow BD = x\) 
\(\tan 30^\circ = \frac{CD}{x} \Rightarrow CD = \frac{x}{\sqrt{3}}\)

Step 2: Find segments

Since AD = \(x\sqrt{3}\): 
\(AB = AD - BD = x\sqrt{3} - x = x(\sqrt{3} - 1)\) 
\(BC = BD - CD = x - \frac{x}{\sqrt{3}} = x\left(\frac{\sqrt{3} - 1}{\sqrt{3}}\right)\) 
\(CD = \frac{x}{\sqrt{3}}\)

Step 3: Ratio

Ratio: \[ AB : BC : CD = (x(\sqrt{3} - 1)) : \Big(x\frac{\sqrt{3} - 1}{\sqrt{3}}\Big) : \frac{x}{\sqrt{3}} \] Cancelling \(x\) and multiplying by \(\sqrt{3}\): \[ (3 - \sqrt{3}) : (\sqrt{3} - 1) : 1 \]

✅ Final Answer:

\( AB : BC : CD = (3 - \sqrt{3}) : (\sqrt{3} - 1) : 1 \)