IMO Practice Test: Circles | Class 9 Mathematics

Question 1 of 6 hard

Two parallel chords of lengths 12 cm and 16 cm are drawn on opposite sides of the center of a circle of radius 10 cm. Find the distance between the two chords.

2 cm

14 cm

10 cm

12 cm

Explanation: Distances from center are $\sqrt{10^2-6^2}=8$ and $\sqrt{10^2-8^2}=6$. Opposites add up: 8 + 6 = 14 cm.

Question 2 of 6 hard

An angle subtended by an arc at the center of a circle is 130°. If a point P is chosen on the remaining major arc, what is the measure of the angle formed at P?

130°

260°

65°

50°

Explanation: The angle subtended by an arc at the center is exactly double the angle subtended at the rim.

Question 3 of 6 hard

ABCD is a cyclic quadrilateral. If the internal bisectors of all four angles are drawn, they cross to form an inner quadrilateral PQRS. This new inner shape is always:

A rhombus

A rectangle

A cyclic quadrilateral

A trapezium

Explanation: Opposite angle sums of the newly formed intersection triangles prove they add up to 180°.

Question 4 of 6 hard

A chord of length $24\sqrt{2}$ cm is placed inside a circle. If the angle subtended by this chord at the center is a right angle (90°), calculate the radius of the circle.

12 cm

24 cm

48 cm

$12\sqrt{2}$ cm

Explanation: Using isosceles right triangle properties at the center: $r^2 + r^2 = (24\sqrt{2})^2 \rightarrow 2r^2 = 1152 \rightarrow r = 24$.

Question 5 of 6 hard

ABCD is a cyclic quadrilateral whose diagonals AC and BD intersect at a right angle (90°) at point P. If a perpendicular line is drawn from P to side BC and extended backwards to meet AD at M, then AM is:

Half of AD

Equal to MD

Double of MD

Perpendicular to AD

Explanation: Angle chasing reveals that triangle PMD and triangle PMA are isosceles, making M the midpoint.

Question 6 of 6 hard

Three friends are standing on a circular boundary ring of radius 5 m. Reshma throws a ball to Salma, Salma to Mandeep, and Mandeep to Reshma. If the distance between Reshma and Salma, and between Salma and Mandeep is 6 m each, find the distance between Reshma and Mandeep.

4.8 m

9.6 m

7.2 m

8.0 m

Explanation: Using area properties of the formed kite layout, half-chord height is 4.8 m, so full chord = 9.6 m.

IMO Practice Test — Circles