Critical Thinking Questions: Circle: Arc and Cyclic Properties & Tangents and Intersecting Chords

Statements Based Questions

Q1: In a circle, if ∠PQR = 45°, where P and R are points on the circle and Q is the centre, what is the measure of the angle subtended by the same arc PR at any point on the remaining part of the circle?
Statement 1: It is 22.5°
Statement 2: It is 45°
Statement 3: It is 90°
(a) Only 1
(b) Only 2
(c) Only 3
(d) None of the above

Q2: Given a circle with centre O and two chords AB and CD that intersect at point E inside the circle. Consider the following statements:
Statement 1: ∠AEB + ∠CED = 180°
Statement 2: ∠AEB = ∠COD
Statement 3: If AB and CD are equal in length, then ∠AEB = ∠CED.
(a) Only 1
(b) Only 2
(c) Only 1 and 2
(d) Only 1 and 3

Q3: For two circles intersecting at points A and B, if AC and AD are the diameters of the two circles respectively, which of the following statements is true?
Statement 1: ∠ABC is a right angle.
Statement 2: ∠ABD is a right angle.
Statement 3: Point B lies on the line segment CD.
(a) Only 1
(b) Only 2
(c) Only 1 and 2
(d) Only 1 and 3

Q4: If in a cyclic quadrilateral ABCD, angle A is 70° and angle C is 110°, then the angles B and D are:
Statement 1: Angle B is 70° and angle D is 110°.
Statement 2: Angle B is 110° and angle D is 70°.
Statement 3: Angle B is 80° and angle D is 100°.
(a) Only 1
(b) Only 2
(c) Only 3
(d) None of the above

Q5: Using the circle theorem, if the diameter of a circle stands on an angle of 45° at a point on the circle, what is the value of the angle that the diameter subtends at the centre of the circle?
Statement 1: The angle at the centre is 45° because the angle at the centre is equal to the angle at the circumference.
Statement 2: The angle at the centre is 90° because the angle at the centre is twice the angle at the circumference.
Statement 3: The angle at the centre is 135° because it is supplementary to the angle at the circumference.
(a) Only 1
(b) Only 2
(c) Only 1 and 3
(d) None of the statements are correct

Q6: Which of the following statements is true regarding a cyclic quadrilateral and its properties?
Statement 1: The sum of the opposite angles of a cyclic quadrilateral is 180°.
Statement 2: If the diagonals of a cyclic quadrilateral bisect each other, the quadrilateral is a rectangle.
Statement 3: In a cyclic quadrilateral, an exterior angle is equal to the interior opposite angle.
(a) Only 1
(b) Only 1 and 3
(c) Only 2
(d) All 1, 2, and 3

Q7: A point P is outside a circle with centre O. Tangents PA and PB are drawn to the circle. Which of the following statements are true?
Statement 1: The lengths of tangents PA and PB from point P to the circle are equal.
Statement 2: The line segment OP is perpendicular to the tangent PA at the point of contact.
Statement 3: The angle between the tangents PA and PB is 90 degrees.
(a) Only 1
(b) Only 2
(c) Only 1 and 2
(d) All 1, 2, and 3

Q8: Given a circle with centre O, chord PQ is parallel to diameter AB, and R is a point on the circle such that PR is a diameter. Which of the following statements are true?
Statement 1: The angle subtended by arc PQ at point R is a right angle.
Statement 2: Chord PQ is equal in length to radius OR.
Statement 3: The angle subtended by arc PR at any point on the remaining part of the circle other than Q is 90°.
(a) Only 1
(b) Only 2
(c) Only 1 and 3
(d) None of the above

Q9: A circle touching the side BC of a triangle ABC at P and is touching AB and AC when produced at Q and R respectively. If AQ = AR and AQ + QR = 2 cm, which of the following are correct?
Statement 1: The perimeter of triangle ABC is 4 cm.
Statement 2: The tangents AP and AQ are equal in length.
Statement 3: The length of tangent QR is 1 cm.
(a) Only 1
(b) Only 2
(c) Only 1 and 3
(d) All 1, 2, and 3

Q10: In a circle with centre O, if ∠BDC = 65° where D lies on the circle, and BC is a tangent to the circle at B, find the angle ∠BOC:
Statement 1: ∠BOC is 130° because it is twice the angle ∠BDC.
Statement 2: ∠BOC is 115° because it is equal to 2 × 65 - 15.
Statement 3: ∠BOC is 155° because the sum of ∠BDC and 90°.
(a) Only 1
(b) Only 2
(c) Only 1 and 3
(d) None of the above