A chord in a circle with radius 10 cm subtends an angle of 90° at the center. What is the area of the minor segment?

Correct — A. Minor segment = sector − the triangle formed by the two radii and the chord.

Sector = (90⁄360) × π × 10² = 25π
Triangle: the radii meet at 90°, so area = ½ × 10 × 10 = 50
Segment = 25π − 50 ≈ 78.5 − 50 ≈ 28.5
So the area is (25π − 50) sq. cm → option (a)

Why the others are wrong

• (b)
  (50π−25) sq. cm — 50π is a 180° sector, twice the 90° one, and 25 is half the triangle. Its value, about 132 sq. cm, exceeds the whole 90° sector (≈ 78.5) that contains the segment.
• (c)
  (50π−50) sq. cm — 50π is the semicircle, not the quarter-circle sector. A 90° angle takes 90⁄360 = ¼ of πr², which is 25π, so the segment is 25π − 50.
• (d)
  (100π−50) sq. cm — 100π is the whole circle. Taking the triangle from it gives neither segment: the major segment is 100π − (25π − 50) = 75π + 50.

A chord splits a circle into two segments. The minor segment lies between the chord and the shorter arc.

Its area is the sector on that arc minus the triangle made by the two radii and the chord: (θ⁄360) × πr² − ½ r² sin θ.

At θ = 90°, sin θ = 1 and the triangle is right-angled, so the segment is r²(π⁄4 − ½).

Key facts

• Area of a sector with central angle θ = (θ⁄360) × πr².
• Area of the triangle formed by two radii with angle θ between them = ½ r² sin θ.
• Minor segment = sector − triangle, and major segment = circle − minor segment.

Common traps

• Using the whole-circle area 100π or the semicircle 50π in place of the 90° sector.
• Taking the chord as the triangle's base and the radius as its height. With a 90° angle at the centre, base and height are the two radii: ½ × 10 × 10.

No directly related past PYQ was found.