IMO Practice Test — Circles
6 Questions • 15 min • Olympiad level
15:00
Question 1 of 6
medium
Two concentric circles have radii of 5 cm and 3 cm respectively. Find the total length of the chord of the larger circle which touches the smaller circle.
4 cm
6 cm
8 cm
10 cm
Explanation: The radius of the inner circle is perpendicular to the chord, bisecting it. Half-chord = sqrt(5 squared - 3 squared) = 4 cm. Full chord = 8 cm.
Question 2 of 6
medium
A right-angled triangle ABC, with sides AB = 6 cm and BC = 8 cm (right-angled at B), has a circle inscribed perfectly inside it. Find the radius of this in-circle.
1 cm
2 cm
2.5 cm
3 cm
Explanation: Hypotenuse AC = 10 cm. In-radius r for a right triangle = (Base + Height - Hypotenuse) / 2 = (6 + 8 - 10) / 2 = 2 cm.
Question 3 of 6
medium
From an external point P, two tangents PA and PB are drawn to a circle centered at O. If angle APB = 120 degrees, determine the exact ratio of length OP to length PA.
1:2
2:1
1:sqrt(3)
sqrt(3):1
Explanation: Line OP bisects angle APB, so angle APO = 60 degrees. In right triangle OAP, cos(60) = PA / OP = 1 / 2. Therefore, OP : PA = 2 : 1.
Question 4 of 6
medium
An isosceles triangle ABC is circumscribed about a circle. If the equal sides are AB = AC = 12 cm, and the base BC = 8 cm, find the length of the side segment from vertex A to the first point of contact.
4 cm
6 cm
8 cm
10 cm
Explanation: Let contact points on AB, AC, BC be D, F, E. Since ABC is isosceles, BE = EC = 4 cm. Thus BD = CF = 4 cm. AD = AB - BD = 12 - 4 = 8 cm.
Question 5 of 6
medium
A circle touches all four sides of a quadrilateral ABCD whose side lengths are expressed algebraically as AB = x + 3, BC = 2x, CD = x + 5, and DA = x + 1. Find the numerical value of x.
3
5
7
9
Explanation: For a circumscribed quadrilateral, AB + CD = BC + DA. (x + 3) + (x + 5) = 2x + (x + 1) => 2x + 8 = 3x + 1 => x = 7.
Question 6 of 6
medium
Two tangents PA and PB are drawn from an external point P to a circle centered at O. If a third tangent line CD is drawn cutting PA at C and PB at D such that it skims the arc AB, and the perimeter of triangle PCD is found to be 30 cm, find the length of PA.
10 cm
15 cm
20 cm
30 cm
Explanation: Perimeter of triangle PCD = PA + PB = 2 * PA (since external tangents CA=CE and DB=DE). 2 * PA = 30 cm, so PA = 15 cm.