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Theorem: tangents from an external point are equal sheet
3-mark answer frame

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

How to lay out the answer. Follow the frame, then write it in your own words.

  1. 1 · Write what is given and what is to be found Given: a circle with centre O, an external point P, and tangents PQ and PR touching it at Q and R. To prove: PQ = PR.
  2. 2 · Draw the figure Circle with centre O; P outside; tangents PQ and PR; join OQ, OR and OP; mark right angles at Q and R.
  3. 3 · Write the formula or theorem The tangent is perpendicular to the radius, so ∠OQP = ∠ORP = 90°.
  4. 4 · Compare the two triangles
    • OQ = OR (radii of the same circle)
    • OP = OP (common)
    • ∠OQP = ∠ORP = 90°
  5. 5 · Conclude ΔOQP ≅ ΔORP (RHS), so PQ = PR (CPCT).
  6. 6 · Check and box the answer Write 'Hence proved'. You may add that ∠OPQ = ∠OPR, so OP bisects ∠QPR.

Scoring tip: Write a reason in brackets beside every equal pair: radii, common, 90°. Examiners mark each reason.