**Math Labs with Activity – Two Arcs of a Circle are Congruent**

**OBJECTIVE**

To verify that if two arcs of a circle are congruent then the corresponding chords are equal

**Materials Required**

- A sheet of white paper
- A piece of cardboard
- A small piece of thread
- A sheet of tracing paper
- A geometry box
- A tube of glue

**Theory**

The theorem can be proved as below.

Consider a circle with centre O and radius r having two equal arcs AB and PQ as shown in Figure 17.1.

Join AO, OB, PO, OQ, AB and PQ.

In ΔAOB and POQ, we have

- OA = OP (each equal to r)
- OB = OQ (each equal to r)
- ∠AOB = ∠POQ (
**∴**two equal arcs subtend equal angles at the centre of the circle)

Then, ΔAOB is congruent to ΔPOQ (by SAS-criterion).

**∴** AB = PQ.

**Procedure**

**Step 1:** Paste the sheet of white paper on the cardboard and mark a point O on this paper. With O as centre, draw a circle with any radius.

**Step 2:** Place the thread on any part of the circle along the circumference. Mark its end points A and B to get an arc AB. Again place the thread on any other part of the circle along the circumference. Mark its end points P and Q to get another arc PQ which is congruent to AB. Join AB and PQ to get the chords corresponding to the arcs AB and PQ as shown in Figure 17.2.

**Step 3:** Trace the circle along with the two chords AB and PQ on the tracing paper.

**Step 4:** Place the tracing paper over Figure 17.1 such that the AB lies over the PQ. Since the two arcs are congruent, AB exactly overlaps PQ.

**Observations**

The chord AB completely overlaps the chord PQ. This shows that the chord AB is equal to the chord PQ.

**Result**

It is verified that if two arcs of a circle are congruent then their corresponding chords are equal.

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