Class 9 Maths Chapter 9 Circles Assertion-Reason Questions

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Class 9 Maths Chapter 9 Circles Assertion-Reason Questions

Assertion-Reason Questions

Directions: In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option: (a) Both A and R are true, and R is the correct explanation of A. (b) Both A and R are true, but R is not the correct explanation of A. (c) A is true, but R is false. (d) A is false, but R is true.

Question 1

Assertion (A): Equal chords of a circle subtend equal angles at the centre. Reason (R): If two chords of a circle subtend equal angles at the centre, then the chords are equal.

Answer: (b)

Explanation: Assertion (A) is true (Theorem 9.1). Reason (R) is also true (Theorem 9.2), which is the converse of Theorem 9.1. Since R is the converse of A, it does not serve as the explanation or proof for why A is true.

Question 2

Assertion (A): If an arc PQ of a circle subtends angle POQ at the centre O and angle PAQ at a point A on the remaining part of the circle, then $\angle POQ = 2 \angle PAQ$. Reason (R): An exterior angle of a triangle is equal to the sum of the two interior opposite angles.

Answer: (a)

Explanation: Assertion (A) states Theorem 9.7. The proof of Theorem 9.7 uses the property that an exterior angle of a triangle (like $\angle BOQ$) is equal to the sum of the two interior opposite angles ($\angle OAQ + \angle AQO$). This property (R), combined with the fact that OA = OQ (radii), is essential to proving the relationship stated in A. Thus, R is the correct explanation of A.

Question 3

Assertion (A): The perpendicular from the centre of a circle to a chord bisects the chord. Reason (R): The length of the perpendicular from a point to a line is defined as the distance of the line from the point.

Answer: (b)

Explanation: Assertion (A) is true (Theorem 9.3). Reason (R) is also a true definition concerning distance in Mathematics. However, the definition of distance (R) is independent of the theorem about chord bisection (A). R does not explain why A is true.

Question 4

Assertion (A): If two chords of a circle are equidistant from the centre, then the chords must be equal in length. Reason (R): Chords equidistant from the centre (or corresponding centres) of a circle (or of congruent circles) are equal.

Answer: (a)

Explanation: Assertion (A) is a statement of Theorem 9.6. Reason (R) is the precise statement of Theorem 9.6, verifying that A is true. Since R is essentially the theorem itself, confirming A’s truth, R is the correct explanation of A.

Question 5

Assertion (A): In a cyclic quadrilateral ABCD, the sum of opposite angles, $\angle A + \angle C$, must be $180^\circ$. Reason (R): If a quadrilateral has the sum of a pair of opposite angles equal to $180^\circ$, it is called a cyclic quadrilateral.

Answer: (b)

Explanation: Assertion (A) is true (Theorem 9.10). Reason (R) is a definition stemming from Theorem 9.11 (the converse of A), stating the condition for a quadrilateral to be cyclic. Both statements are true, but R (the sufficient condition for being cyclic) does not explain the necessity stated in A (the property of being cyclic).

Question 6

Assertion (A): If a diameter AD of a circle subtends an angle at a point B on the circumference, then $\angle ABD = 90^\circ$. Reason (R): The angle in a semicircle is a right angle.

Answer: (a)

Explanation: A diameter forms a semicircle. The angle subtended by a diameter at any point on the circle is an angle in a semicircle. Assertion (A) is a specific instance of the property stated in Reason (R). Therefore, R is the correct explanation of A.

1. What is the relationship between two chords of a circle if the angles they subtend at the centre are equal?
  • (A) The chords are perpendicular to each other
  • (B) The chords are parallel to each other
  • (C) The chords are equal in length
  • (D) The chords are equidistant from the circumference
    • Answer

    Answer: (C) The chords are equal in length

     2. The perpendicular from the centre of a circle to a chord performs which action?
  • (A) It doubles the length of the chord
  • (B) It bisects the chord
  • (C) It subtends a $60^\circ$ angle at the center
  • (D) It must pass through a point on the minor arc
    • Answer

    Answer: (B) It bisects the chord

     3. If two chords of a circle are equal, what can be concluded about their distance from the centre?
  • (A) They are equidistant from the centre
  • (B) The longer chord is farther from the centre
  • (C) Their distances sum to the radius
  • (D) Their distances are unequal
    • Answer

    Answer: (A) They are equidistant from the centre

     4. If an arc subtends an angle of $100^\circ$ at the centre of a circle, what angle does the same arc subtend at a point on the remaining part of the circle?
  • (A) 200°
  • (B) 100°
  • (C) 50°
  • (D) 90°
    • Answer

    Answer: (C) 50°

     5. What is the specific property of the sum of either pair of opposite angles of a cyclic quadrilateral?
  • (A) The sum is always $90^\circ$
  • (B) The sum is always $180^\circ$
  • (C) The sum is always $360^\circ$
  • (D) The sum must be less than $180^\circ$
    • Answer

    Answer: (B) The sum is always $180^\circ$

     6. What is the measure of the angle formed in a semicircle?
  • (A) 45°
  • (B) 180°
  • (C) 90°
  • (D) 360°
    • Answer

    Answer: (C) 90°

     7. If a line segment joining two points subtends equal angles at two other points lying on the same side of the line, what conclusion can be drawn about the four points?
  • (A) They form a trapezium
  • (B) They are concyclic (lie on a circle)
  • (C) They form a right triangle
  • (D) The points must be vertices of a congruent quadrilateral
    • Answer

    Answer: (B) They are concyclic (lie on a circle)

     8. When are two circles defined as congruent?
  • (A) If they have different diameters
  • (B) If they have the same centre
  • (C) If they have the same area
  • (D) If they have the same radii
    • Answer

    Answer: (D) If they have the same radii

     9. A quadrilateral ABCD is called cyclic if:
  • (A) Only two vertices lie on a circle
  • (B) All four vertices lie on a circle
  • (C) Its diagonals are equal
  • (D) Its opposite sides are equal
    • Answer

    Answer: (B) All four vertices lie on a circle

     10. Which relationship is true between the length of a chord and its distance from the centre?
  • (A) Longer chords are nearer to the centre than smaller chords
  • (B) Shorter chords are nearer to the centre than longer chords
  • (C) All chords are equidistant from the centre
  • (D) Only perpendicular chords have defined distances from the centre
    • Answer

    Answer: (A) Longer chords are nearer to the centre than smaller chords
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