Math Labs with Activity - Angles Subtended by Two Chords

Math Labs with Activity – Angles Subtended by Two Chords

OBJECTIVE

To verify that if the angles subtended by two chords at the centre of a circle are equal then the chords are equal

Materials Required

1. A sheet of white paper
2. A piece of cardboard
3. A sheet of tracing paper
4. A geometry box
5. A tube of glue

Theorem
The theorem to be verified is the converse of the theorem verified in Activity 15.
The theorem can be proved as below.
Consider a circle with radius r and centre O having two chords AB and PQ such that ∠AOB = ∠POQ (see Figure 16.1).

In ΔAOB and POQ, we have

1. AO = OP (each equal to r)
2. BO = OQ (each equal to r)
3. ∠AOB = ∠POQ (given)

Then, ΔAOB is congruent to ΔPOQ (by SSA-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: Draw two equal angles at the centre O of the circle.
(This may be done using a pair of compasses as you have already learnt in the lower classes). Let these two angles be subtended by arcs Fig. 16.2.
AB and PQ as shown in Figure 16.2. Join AB and PQ.

Step 3: Trace the ΔAOB on the tracing paper.
Step 4: Place the ΔAOB obtained on the tracing paper over the ΔPOQ such that ∠AOB is superimposed over ∠POQ.

Observations
We observe that ΔAOB completely overlaps ΔPOQ. Therefore, ΔAOB is congruent to ΔPOQ.
So, we conclude that the chord AB is equal to the chord PQ.

Result
It is verified that if the angles subtended by two chords at the centre of a circle are equal then the chords are equal.

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