A pole 10 m long rests slantly against a vertical wall AB making an angle 30° with the horizontal (ground). Find how far the foot of the pole is from the wall (in metres).

Correct — A. The four choices are images, reading 5√3, 10, 15 and 5⁄√3 in order.

The pole, the wall and the ground form a right triangle, with the right angle at the foot of the wall.

The pole is the hypotenuse = 10 m, and the 30° sits between the pole and the ground.
The distance from the foot of the pole to the wall is therefore the side adjacent to the 30°.

adjacent = hypotenuse × cos 30°
= 10 × (√3⁄2) = 5√3 m → option (a).

Check: 5√3 ≈ 8.66 m, shorter than the 10 m pole, as any leg must be.

Why the others are wrong

• (b)
  10 is the pole itself, the hypotenuse. The gap between the pole's foot and the wall is a leg of the right triangle, so it has to come out below 10 m.
• (c)
  15 is longer than the 10 m pole, which no leg of the triangle can be. It is 10 × 3⁄2, i.e. cos 30° misremembered as 3⁄2 instead of √3⁄2.
• (d)
  5⁄√3 is about 2.89 m and comes from 10 ÷ (2√3) — inverting the ratio. Since cos 30° = √3⁄2, the adjacent side is 10 multiplied by √3⁄2.

Every leaning-pole or ladder question is one right triangle with three labelled parts: the pole is always the hypotenuse, the wall is the vertical leg and the gap along the ground is the horizontal leg.

The only decision is which leg the given angle names. Here the angle is measured with the horizontal, so the ground gap is adjacent to it and the height on the wall is opposite. Adjacent over hypotenuse is cosine, which settles the ratio before any arithmetic starts.

Key facts

• cos 30° = √3⁄2, so 10 × cos 30° = 5√3 ≈ 8.66.
• The hypotenuse is the longest side of a right triangle, so both legs here must be under 10 m.
• Measuring the angle from the horizontal makes the ground distance the adjacent side and the wall height the opposite side.
• The height reached on the wall in the same triangle is 10 × sin 30° = 5 m.

Common traps

• Using sin 30° and answering 5, which is the height on the wall rather than the distance from it.
• Reading the 30° as the angle between the pole and the wall instead of the ground.
• Converting 5√3 to 8.66 when every option is written in surd form.

No directly related past PYQ was found.