A taut wire supports a vertical pole. The wire is fixed to the ground 10 meters away from the base of the pole and makes an angle of 60° with the ground. What is the length of the wire?

Correct — C. The pole, the ground and the wire make a right triangle, with the right angle at the foot of the pole.

Ground distance, the side next to the 60° angle: 10 m
The wire is the hypotenuse, so use cos 60° = adjacent ⁄ hypotenuse

cos 60° = 10 ⁄ L
1⁄2 = 10 ⁄ L
Cross-multiply: L = 2 × 10 = 20 m → option (c)

Why the others are wrong

• (a)
  10 m — 10 m is the ground distance, one leg of the triangle. The wire is the hypotenuse, which is longer than either leg.
• (b)
  10√3 m — 10√3 m is the height at which the wire meets the pole: tan 60° × 10 = 10√3 ≈ 17.3 m. That is the vertical leg, not the wire.
• (d)
  20√3 m — 20√3 m ≈ 34.6 m would reach the ground 20√3 × cos 60° = 10√3 ≈ 17.3 m from the pole, not the 10 m given.

In a 30°-60°-90° triangle the sides opposite 30°, 60° and 90° are in the ratio 1 : √3 : 2.

The angle where the wire meets the pole is 180° − 90° − 60° = 30°. The 10 m along the ground is opposite it, so the wire is 2 × 10 = 20 m and the point on the pole is 10√3 m up.

Key facts

• sin 60° = √3⁄2, cos 60° = 1⁄2, tan 60° = √3.
• In a 30°-60°-90° triangle the sides are 1 : √3 : 2, opposite 30°, 60° and 90°.
• Given the side adjacent to an angle θ, hypotenuse = adjacent ÷ cos θ.

Common traps

• Using tan 60° and stopping at 10√3 m, which is the height on the pole, not the wire.
• Writing sin 60° = 10 ⁄ L, which puts the 10 m opposite the 60° angle and gives L = 20⁄√3 ≈ 11.5 m.

No directly related past PYQ was found.