Multiple choice

A chord of a circle of radius $7$ cm. subtends an angle of $90^o$ at its centre. The ratio of areas of smaller and larger segment is

  1. $2:7$
  2. $1:10$
  3. $1:11$

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Area of circle = pi * 7^2 = 49 * pi. Area of sector = (90/360) * 49 * pi = 12.25 * pi. Area of triangle = (1/2) * 7 * 7 = 24.5. Smaller segment = 12.25 * pi - 24.5 = 12.25 * (3.14 - 2) = 13.96. Larger segment = 49 * pi - 13.96 = 139.86. The ratio is approximately 1:10.

AI explanation

The area of the sector for a 90 degree angle at the center is one-fourth of the circle's area, which is one-fourth of pi times 7 squared, or 38.5. The area of the right triangle formed by the radii is half of 7 times 7, giving 24.5. The smaller segment area is the sector area minus the triangle area, giving 14, and the larger segment is the rest of the circle, which is 154 minus 14, equaling 140. The ratio of 14 to 140 simplifies exactly to 1 to 10.

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