CAT 2022 Slot 3 QA Q18 Geometry

Question 18 of the Quantitative Ability (QA) section, CAT 2022 Slot 3, with CAT's official answer and a worked solution.

In a triangle \(\mathrm{ABC}, \mathrm{AB}=\mathrm{AC}=8 \mathrm{cm}\). A circle drawn with \(\mathrm{BC}\) as diameter passes through \(\mathrm{A}\). Another circle drawn with center at A passes through B and \(\mathrm{C}\). Then the area, in sq. \(\mathrm{cm}\), of the overlapping region between the two circles is
  1. A
    \(16(\pi-1)\)
  2. B
    \(32(\pi-1)\)
  3. C
    \(32 \pi\)
  4. D
    \(16 \pi\)

Answer: (B) 32(π-1) Geometry MCQ

Solution

  1. A lies on the circle with diameter BC, so \(\angle A=90^\circ\), \(BC=8\sqrt2\) and that circle has radius \(4\sqrt2\).
  2. The second circle has centre A and radius 8. The overlap is the half of the small circle on A's side of BC plus the segment of the big circle beyond chord BC.
  3. Half of small circle: \(\frac12\pi(4\sqrt2)^2=16\pi\).
  4. Segment: quarter-circle sector of radius 8 minus triangle ABC \(=16\pi-32\).
  5. Overlap \(=32\pi-32=32(\pi-1)\).

Same concept in other years

See every question on Circular segment area = sector − triangle.

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