Question

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

Hint:

A neat diagram showing the circle, tangents, and radii is essential for full marks in geometry proofs.

Answer 1

Theorem:
The lengths of tangents drawn from an external point to a circle are equal.

Given:
A circle with centre O.
From an external point P, two tangents PA and PB are drawn to the circle, touching it at points A and B respectively.

To Prove:
\[ PA = PB \]

Step 1: Draw radii to the points of contact
Join OA and OB.
Since a radius drawn to the point of contact of a tangent is perpendicular to the tangent:
\[ OA \perp PA,\quad OB \perp PB \]
Therefore, ΔOPA and ΔOPB are right triangles.

Step 2: Compare right triangles ΔOPA and ΔOPB
In the two triangles:
• OP = OP (common side)
• OA = OB (radii of the same circle)
• \(\angle OAP = \angle OBP = 90^\circ\)

So, by RHS (Right angle–Hypotenuse–Side) congruence rule:
\[ \triangle OPA \cong \triangle OPB \]

Step 3: Conclude equality of tangent lengths
From congruent triangles:
\[ PA = PB \]

Final Result:
The lengths of tangents drawn from an external point to a circle are equal.

Hence proved.