**Math Labs with Activity – The Lengths of the Tangents Drawn from an External Point to a Circle**

**OBJECTIVE**

To verify that the lengths of the tangents drawn from an external point to a circle are equal

**Materials Required**

- A sheet of transparent paper
- A geometry box

**Theory**

The theorem can be proved as follows.

Let two tangents AP and AQ be drawn from a point A (external point) to a circle with its centre at O and having a radius r.

Join OP, OQ and OA.

In AOPA and OQA, we have

- OP=OQ (each is equal to r)
- OA =OA (common)
- ∠OPA = ∠OQA (each is equal to 90° since AP and AQ are tangents).

**∴** ΔOPA is congruent to ΔOQA (by RHS-criterion).

Hence, AP = AQ, i.e., the two tangents are equal.

**Procedure**

**Step 1:** Mark a point O on the sheet of transparent paper. With O as the centre, draw a circle of any radius.

**Step 2:** Mark a point A outside the circle.

**Step 3:** Fold the paper along the line that passes through the point A and just touches the circle. Make a crease and unfold the paper. Mark the point P where the line of fold touches the circle. Join AP. Then, AP is one of the tangents to the circle from the point A.

**Step 4:** Fold the paper along the line that passes through the point A and just touches the circle at a point other than P. Make a crease and unfold the paper. Mark the point Q where the line of fold touches the circle. Join AQ. Then, AQ is another tangent to the circle through the point A.

**Step 5:** Join OP, OQ and OA, as shown in Figure 30.2.

**Step 6:** Fold the paper along the line OA.

**Observations**

We observe that when the paper is folded along the line OA, the point P falls exactly on the point Q. Therefore, AP = AQ.

**Result**

It is verified that the lengths of the tangents drawn from an external point to a circle are equal.

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