Arc Properties of a Circle — Assertion-Reason Type Questions

Assertion (A): In the figure, two congruent circles have centres O and O′.

Arc AXB subtends an angle of 60° at the centre O and arc AYB′ subtends an angle of 20° at the centre O′.

Then the ratio of arcs AXB and AY′B′ is 3 : 1.

Reason (R): Congruent arcs of a circle subtend equal angles at the centre.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

Given, two circles are congruent.

For congruent circles,

Length of an arc is proportional to angle subtended at the centre.

$\dfrac{Arc AXB}{arc AY′B′} = \dfrac{60°}{20°} = \dfrac{3}{1}$

∴ Assertion (A) is true.

Congruent arcs of a circle subtend equal angles at the centre because the length of an arc is directly proportional to the angle subtended by it at the centre.

∴ Reason (R) is true.

Hence, Option 3 is the correct option.

Assertion (A): Two congruent circles with centre O and O′ intersect at two points A and B. Then ∠AOB = ∠AO′B.

Reason (R): If a pair of opposite sides of a cyclic quadrilateral are equal, then its diagonals bisect each other.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

Join AB, OA, OB, O’A and O’B.

In triangle AOB and triangle AO’B,

⇒ OA = AO’ (circles have same radius as they are congruent)

⇒ OB = BO’ (circles have same radius as they are congruent)

⇒ AB = AB (common chord)

From the SSS congruence criterion,

△ AOB ≅ △ AO’B

Since, corresponding angles of congruent triangle are equal.

⇒ ∠AOB = ∠AO’B

∴ Assertion (A) is true.

If a pair of opposite sides of a cyclic quadrilateral are equal, then its diagonals bisect each other is not always true.

∴ Reason (R) is false.

Hence, Option 1 is the correct option.