If arcs PAQ and RBS of a circle are congruent, then find the ratio of PQ and RS.

Correct — C. Congruent arcs of one circle are equal in length, so they subtend equal angles at the centre.

Rule: equal arcs ⇒ equal central angles ⇒ equal chords

Let O be the centre. Arc PAQ ≅ arc RBS gives ∠POQ = ∠ROS.
In ∆POQ and ∆ROS the sides OP, OQ, OR, OS are all the radius, and the included angles are equal.
So the two triangles are congruent by SAS, which forces PQ = RS.

PQ : RS = 1 : 1 → option (c).

Why the others are wrong

• (a)
  1 : 3 — 1 : 3 makes RS three times PQ. Congruence is equality, not proportion, so no multiple other than 1 can appear.
• (b)
  2 : 3 — 2 : 3 is what unequal chords look like, and unequal chords need different central angles. Congruent arcs fix those angles equal.
• (d)
  1 : 2 — 1 : 2 tempts because doubling an arc feels like doubling its chord. It does not — chord length is 2r sin(θ⁄2), which grows more slowly than θ.

Inside one circle three quantities move together: arc length, central angle and chord. Fix any one of them and the other two are fixed with it.

The chain runs arc = rθ, so equal arcs give equal θ. Equal θ with equal radii gives congruent triangles by SAS, and congruent triangles give equal chords.

The converse holds as well — equal chords in one circle cut off equal minor arcs — which is what lets the question be asked from either end.

Key facts

• Congruent arcs of the same circle subtend equal angles at the centre.
• Equal central angles in a circle of fixed radius give equal chords.
• A chord subtending a central angle θ in a circle of radius r has length 2r sin(θ⁄2).

Common traps

• Reading congruent as similar and hunting for a ratio other than 1 : 1
• Assuming chord length is proportional to arc length, which the sine relation denies

No directly related past PYQ was found.