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      The tangent at any point on a circle is perpendicular to the radius at that point

      Lesson details

      Learning outcome

      I can derive and use the theorem: the tangent at any point on a circle is perpendicular to the radius at that point.

      Key learning points

      1. A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments
      2. Theorems can be thought of as puzzles to solve, you are showing how to find a result
      3. In order to use this theorem, you may need to draw a diagram or add more information to an existing one

      Keywords

      • Tangent - A tangent of a circle is a line that intersects the circle exactly once.

      • Chord - A chord is any line segment joining two points on the circumference of a circle.

      • Perpendicular - Two lines are perpendicular if they meet at right angles.

      Common misconception

      Pupils may assume that tangents can be drawn in slightly different directions at the same point on a circle.

      The tangent at a point on a circle can only be drawn in one possible direction. The tangent is always perpendicular to the radius at that point.

      Teacher tip

      The lesson demonstrates how this circle theorem could be explored practically by using a circular geoboard and using dynamic geometry software. Pupils may wish to explore these further for themselves.

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
      except where otherwise stated. See Oak's terms & conditions
      (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

      6 Questions

      Q1.
      The is a line which intersects the circle exactly once.

      Correct Answer: tangent, tangent of a circle, circle tangent

      Q2.
      The size of the angle labelled $$x$$° is °.

      An image in a quiz
      Correct Answer: 26, 26°, 26 degrees

      Q3.
      MY = 8 cm, OZ = 13 cm, ZM = 12 cm, OM = 5 cm. Find the distance MX.

      An image in a quiz
      5 cm
      8 cm
      Correct answer: 12 cm
      13 cm
      Not enough information.

      Q4.
      The size of the angle labelled $$m$$° is ° (in degrees).

      An image in a quiz
      Correct Answer: 71, 71°, 71 degrees

      Q5.
      The size of the angle labelled $$a$$° is °.

      An image in a quiz
      Correct Answer: 46, 46°, 46 degrees

      Q6.
      The angle subtended from arc AB at the circumference of the circle is 11°. Find ∠BOA (in degrees).

      An image in a quiz
      5.5°
      11°
      Correct answer: 22°
      79°
      169°

      6 Questions

      Q1.
      The size of the angle labelled $$x$$° on this 9-point geoboard is °.

      An image in a quiz
      Correct Answer: 80, 80°, 80 degrees, 80degrees

      Q2.
      Find ∠OAB on this 9-point geoboard (in degrees).

      An image in a quiz
      40°
      Correct answer: 50°
      80°
      90°
      Not enough information.

      Q3.
      Find ∠EFO on this 8-point geoboard (in degrees).

      An image in a quiz
      Correct Answer: 90, 90°, 90 degrees

      Q4.
      Find ∠DEF on this 8-point geoboard (in degrees).

      An image in a quiz
      Correct Answer: 45, 45°, 45 degrees, 45degrees

      Q5.
      Find the size of angle $$x$$° (in degrees).

      An image in a quiz
      Correct Answer: 13, 13°, 13 degrees

      Q6.
      Find the size of angle $$y$$° (in degrees).

      An image in a quiz
      Correct Answer: 79, 79°, 79 degrees

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