Question

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

Hint:

Tangents drawn from an external point to a circle are equal in length — always prove this using RHS congruence.

Answer 1

Step 1: Construction and given information.
Let \( P \) be an external point from which two tangents \( PA \) and \( PB \) are drawn to a circle with centre \( O \). We need to prove that \( PA = PB \).

Step 2: Join \( OA \) and \( OB \).
Since \( OA \) and \( OB \) are radii, and \( PA \) and \( PB \) are tangents, \[ OA \perp PA \quad \text{and} \quad OB \perp PB \]
Step 3: Consider triangles \( \triangle OAP \) and \( \triangle OBP \).
In both triangles: \[ OA = OB \quad \text{(radii of the same circle)} \] \[ OP = OP \quad \text{(common side)} \] \[ \angle OAP = \angle OBP = 90^\circ \]
Step 4: Apply RHS congruence criterion.
By the RHS congruence rule, \[ \triangle OAP \cong \triangle OBP \] Step 5: Conclusion.
Hence, \( PA = PB \) (corresponding sides of congruent triangles). \[ \boxed{PA = PB} \]