261. Equal Arcs, Equal Angles · The central angle dictates the arc length

Try to break it

Drag A, B, C, and D around the circle. When the central angle ∠AOB equals ∠COD, the two arcs APB and CQD come out the same length. Pull ∠AOB larger than ∠COD and arc APB stretches longer too; pull it smaller and the arc shrinks. Equal angles at the centre ⇔ equal arcs — one always tracks the other.

How you build it

Draw a circle, centre O.

• Point tool: mark the centre O near the middle.
• Circle tool: click O, then click outward to set the radius.
• Point tool: click on the circle to place A.
• Point tool: click on the circle to place B. The angle ∠AOB can be any size — this is your first central angle.
• Segment tool: join O to A.
• Segment tool: join O to B.
• Segment tool: join A to B — the chord of the first central angle ∠AOB.
• Point tool: click on the circle to place C, away from A and B.
• Arc tool: click A, then B — this sets the compass to the exact length of chord AB.
• Arc tool: centre C (same width) — the arc crosses the circle where D belongs, so that CD = AB.
• Point tool: mark D on the circle where the arc crosses it — now CD = AB.
• Segment tool: join O to C.
• Segment tool: join O to D.
• Segment tool: join C to D. Because CD = AB, the central angle ∠COD equals ∠AOB — equal chords cut off equal arcs.

The proof, step by step

Prove that equal central angles of a circle cut off equal arcs.

• OA = OC (Radii of the same circle)
• OB = OD (Radii of the same circle)
• ∠AOB = ∠COD (Given)
• ΔAOB ≅ ΔCOD (SAS Congruence)
• AB = CD (CPCTC)
• arc APB = arc CQD (Equal chords cut equal arcs)

Worked example

In a circle of radius 10 cm, two chords subtend angles of 60° and 90° at the centre. Which chord is longer?

The chord subtending 90° is longer. In a circle, a larger central angle corresponds to a longer chord (and a larger arc).

• The chord subtending 60°
• The chord subtending 90° — correct
• Both chords are equal
• Cannot be determined