Asked in: March 2025 and March 2026 board papersProve that: The tangent segments drawn from an external point to a circle are congruent.

Answer: AP ≅ AQ (proved below)

[Figure: circle with centre O; from external point A, tangents touch the circle at P and Q; radii OP, OQ and seg OA are drawn.]

Given: O is the centre of the circle; AP and AQ are tangent segments from external point A, touching the circle at P and Q.

To prove: seg AP ≅ seg AQ. Construction: Draw seg OP, seg OQ and seg OA.

Proof: \(\angle OPA = \angle OQA = 90^\circ\) (tangent theorem: a tangent is perpendicular to the radius at the point of contact).

In right-angled \(\Delta OPA\) and \(\Delta OQA\): seg OP ≅ seg OQ (radii of the same circle) and seg OA ≅ seg OA (common side, hypotenuse).

\(\therefore \Delta OPA \cong \Delta OQA\) (hypotenuse-side test) \(\therefore\) seg AP ≅ seg AQ (c.s.c.t.). Hence proved.