2023 · Q82
Consider the following statements:
- 1The angle in a sector greater than a semi-circle is less than a right angle.
- 2If two sides of a pair of opposite sides of a cyclic quadrilateral are equal, then its diagonals are also equal.
Which of the statements given above is/are correct?
Elementary MathematicsCircles
- A1 only
- B2 only
- CBoth 1 and 2
- DNeither 1 nor 2
Explanation
Statement 1 is false: a sector larger than a semi-circle subtends a central angle exceeding 180°, which is certainly not less than a right angle (90°). Statement 2 is true: in a cyclic quadrilateral ABCD with opposite sides AB = CD, the equal chords subtend equal angles at the circumference, giving ∠ABC = ∠DCB; with BC common, △ABC ≅ △DCB by SAS, so the diagonals AC and BD are equal. Only statement 2 is correct, so the answer is (B).