**Math Labs with Activity – Equal Chords of a Circle Subtend**

**OBJECTIVE**

To verify that equal chords of a circle subtend equal angles at the centre of the circle

**Materials Required**

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

**Theory**

A line segment joining any two points on a circle is called a chord of the circle. Any two equal chords of a circle subtend equal angles at the centre of the circle.

The theorem can be proved as below.

Consider a circle with radius r and centre O and having two equal chords AB and PQ as shown in Figure 15.1.

In ΔAOB and POQ, we have

- AO=OP (each equal to r)
- BO =OQ (each equal to r)
- AB = PQ (equal chords)

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

**∴** ∠AOB = ∠POQ.

**Procedure**

**Step 1:** Paste the white paper on the cardboard and draw a circle with centre O on this paper.

**Step 2:** Take a pair of compasses. Placing its needle point at a point A on the circle and taking any radius, draw an arc cutting the circle at some point B. Joining AB we get a chord of the circle.

Taking the same radius and again placing the needle point of the compasses at another point P on the circle, draw an arc cutting the circle at some point Q. Joining PQ we get another chord which is equal to the chord AB.

**Step 3:** Join AO and BO to form the triangle AOB. Also, join PO and QO to form the triangle POQ (see Figure 15.2).

**Step 4:** Trace the ΔAOB on the tracing paper.

**Step 5:** Place the ΔAOB obtained on the tracing paper over the ΔPOQ such that AB overlaps PQ.

**Observations**

We observe that the ΔAOB completely overlaps the ΔPOQ. Therefore, the ΔAOB is congruent to the ΔPOQ.

So, we conclude that ∠AOB = ∠POQ.

**Result**

It is verified that equal chords of a circle subtend equal angles at the centre of the circle.

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