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Theorem: the tangent is perpendicular to the radius sheet
3-mark answer frame

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

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 and a tangent XY at the point P. To prove: OP ⟂ XY.
  2. 2 · Draw the figure Circle with centre O; tangent XY touching it at P; any point Q on XY; join OQ, cutting the circle at R.
  3. 3 · Construction Take a point Q on XY other than P and join OQ.
  4. 4 · Show that Q lies outside If Q were inside or on the circle, XY would meet the circle twice and be a secant. So Q is outside, and OQ meets the circle at R.
  5. 5 · Compare the lengths OQ = OR + RQ > OR, and OR = OP (radii). So OQ > OP for every Q ≠ P.
  6. 6 · Conclude OP is the shortest distance from O to XY. The shortest segment from a point to a line is perpendicular to it. So OP ⟂ XY.
  7. 7 · Check and box the answer Write 'Hence proved' and box it.

Scoring tip: Write Given, To prove and Construction as separate headings. In a 3-mark proof the figure and these headings usually earn the first mark.