Math Labs with Activity - Two Arcs of a Circle are Congruent

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

1. A sheet of white paper
2. A piece of cardboard
3. A small piece of thread
4. A sheet of tracing paper
5. A geometry box
6. 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

1. OA = OP (each equal to r)
2. OB = OQ (each equal to r)
3. ∠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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