From a point A on the ground, the angle of elevation to the top of a building is 30°. From a point B, which is directly between A and the foot of the building, the angle of elevation is 60°. What is the ratio of the distance of point A from the building's foot to the distance of point B from the building's foot?

Correct — D.
Call the building's height h and its foot F.
From A: tan 30° = h ÷ AF, so AF = h ÷ (1⁄√3) = h√3
From B: tan 60° = h ÷ BF, so BF = h ÷ √3 = h⁄√3

Divide: AF ÷ BF = h√3 ÷ (h⁄√3) = √3 × √3 = 3
So AF : BF = 3 : 1 → option (d)

Why the others are wrong

• (a)
  1 : 3 — 1 : 3 is the ratio turned round, B's distance to A's. A sees the top at the smaller angle, so A is farther away and its distance is the larger term.
• (b)
  √3 : 1 — √3 : 1 compares AF with the height, h√3 : h, not with BF. Dividing by BF = h⁄√3 brings in a second √3, making the ratio 3 : 1.
• (c)
  2 : 1 — 2 : 1 is AB : BF, the gap between the two points against B's distance. With BF = 1 unit and AF = 3 units, AB = 2 units, but the question asks for AF : BF.

Distance from the foot = height ÷ tan (angle of elevation). Both points look at the same height, so their distances are in the inverse ratio of the tangents.

tan 30° = 1⁄√3 and tan 60° = √3, so tan 60° is three times tan 30°. The point that sees the smaller angle is three times as far away.

The height never needs a value: it cancels in the ratio.

Key facts

• tan 30° = 1⁄√3 and tan 60° = √3
• Horizontal distance from the foot = height ÷ tan (angle of elevation)
• For one height seen at 30° and at 60°, the distances from the foot are in the ratio 3 : 1

Common traps

• Writing B's distance first (1 : 3) when the question names A's distance first
• Taking AB : BF = 2 : 1, the gap against B's distance, for AF : BF
• Assuming the distance halves because the angle doubles from 30° to 60°

No directly related past PYQ was found.