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Assertion (A): In the given figure the lengths of arc AB and arc BC are in the ratio 2:1. If ∠AOB = 96°, then ∠AOC = 144°.

In the given figure the lengths of arc AB and arc BC are in the ratio 2:1. If ∠AOB = 96°, then ∠AOC = 144°. Assertion Reasoning, Concise Mathematics Solutions ICSE Class 9.

Reason (R): In two equal (congruent) circles if two arcs are equal, then they subtend equal angles at their centres.

  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.

Circles

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Answer

Both A and R are true.

Explanation

Given,

Arc AB : Arc BC = 2:1.

And, ∠AOB = 96°

To find : ∠AOC

Let x be the central angle of arc BC. Then, the central angle of arc AB is 2x.

So, 2x = 96°

⇒ x = 96°2\dfrac{96°}{2}

⇒ x = 48°

∠ BOC = 48°

⇒ ∠ AOC = ∠ AOB + ∠ BOC

= 96° + 48°

= 144°

∴ Assertion (A) is true.

In two equal (congruent) circles if two arcs are equal, then they subtend equal angles at their centres. Assertion Reasoning, Concise Mathematics Solutions ICSE Class 9.

Given : Two circles with center O and O'. Arcs APB and CQD are equal in length.

To Prove : ∠ AOB = ∠ COD.

Construction : Draw chords AB and CD.

Proof : Since, two circles are congruent then equal arcs of a circle cut equal chords.

∴ arc APB = arc CQD

⇒ chord AB = chord CD

In Δ AOB and Δ COD,

OA = OC (Radii of the same circle)

OB = OD (Radii of the same circle)

AB = CD (Proved above)

By SSS Congruency Criterion,

Δ AOB ≅ Δ COD

By corresponding parts of congruent triangles,

∠ AOB = ∠ COD

∴ Reason (R) is true.

Hence, both Assertion (A) and Reason (R) are true.

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