What is the area of the segment formed by a chord in a circle of radius 12 cm, if the angle subtended at the center is 150°?

Correct — A. Segment = sector − triangle, where the triangle joins the centre to the ends of the chord.

Sector: (150⁄360) × π × 12² = (5⁄12) × 144π = 60π
Triangle: ½ × 12 × 12 × sin 150° = 72 × ½ = 36
Segment: 60π − 36 → option (a)

Why the others are wrong

• (b)
  36π−36 — 36π is a 90° sector, a quarter of the circle's 144π. The angle here is 150°, which is 5⁄12 of the circle, so the sector is 60π.
• (c)
  60π−72 — 72 is ½ × 12 × 12 with sin 150° left out. The triangle needs that factor, 72 × ½ = 36, so the segment is 60π − 36.
• (d)
  36π−72 — 36π − 72 makes both slips: a 90° sector instead of 150°, and a triangle of 72 with sin 150° = ½ dropped.

A chord cuts a circle into two segments. The minor segment lies between the chord and the shorter arc, and its area is the sector minus the triangle formed by the two radii and the chord.

Sector area = (θ⁄360) × πr². Triangle area = ½ r² sin θ.

For angles above 90°, use sin θ = sin(180° − θ), so sin 150° = sin 30° = ½.

Key facts

• Minor segment area = (θ⁄360)πr² − ½ r² sin θ.
• sin 150° = sin 30° = ½.
• A 150° sector is 5⁄12 of the whole circle.

Common traps

• Using ½ r² as the triangle's area. That holds only at 90°, where sin θ = 1.
• Using sin 150° = √3⁄2, which is the value of sin 120°.

No directly related past PYQ was found.