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      Identifying which circle theorem to use

      Lesson details

      Learning outcome

      I can analyse a circle problem and identify and use an appropriate circle theorem to solve it.

      Key learning points

      1. There are many circle theorems so it is important to be clear about each one
      2. Extra lines can be added to a diagram to help determine which theorem to use
      3. Some problems require the use of more than one theorem

      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.

      • Equidistant - Points A and B are equidistant from a third point C if the distance AC is equal to the distance BC.

      Common misconception

      Pupils may conflate cyclic quadrilaterals with cases of "the angle at the centre of a circle is twice the angle at the circumference" theorem, when the angle at the centre is a reflex angle.

      While the two diagrams may look similar at a glance, pay extra attention to whether all four points of the quadrilateral are on the circumference or if one of the points is in the centre.

      Teacher tip

      For some questions, different combinations of circle theorems can be applied to solve the problem. Encourage pupils to write down which circle theorem they are using at each stage of their solution and discuss the efficiency of alternate solutions.

      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.
      ∠AKC = 21° ∠AWK is subtended by the chord AK. $$a$$° = °.

      An image in a quiz
      Correct Answer: 21, 21°, 21 degrees

      Q2.
      The image shows a circle, a tangent to the circle and three line segments forming a triangle inside the circle. The size of the angle labelled $$b$$° is °.

      An image in a quiz
      Correct Answer: 56, 56°, 56 degrees

      Q3.
      Find the size of angle $$c$$° (in degrees).

      An image in a quiz
      Correct Answer: 69, 69°, 69 degrees

      Q4.
      The size of angle $$d$$° is °.

      An image in a quiz
      Correct Answer: 124, 124°, 124 degrees

      Q5.
      The size of angle $$e$$° is ° .

      An image in a quiz
      Correct Answer: 57, 57°, 57 degrees

      Q6.
      Calculate the size of angle $$g$$° (in degrees).

      An image in a quiz
      28.5°
      33°
      Correct answer: 114°
      147°
      Not enough information

      6 Questions

      Q1.
      Points X and Y are points where two separate tangents intersect the circle. $$a$$° = °.

      An image in a quiz
      Correct Answer: 222, 222°, 222 degrees

      Q2.
      Points X and Y are points where two separate tangents intersect the circle. Calculate the size of the angle labelled $$b$$°.

      An image in a quiz
      42°
      48°
      Correct answer: 69°
      111°
      Not enough information.

      Q3.
      Points A and B are points where two separate tangents intersect the circle. Calculate the size of the angle labelled $$c$$°.

      An image in a quiz
      38°
      41°
      45°
      Correct answer: 49°
      Not enough information.

      Q4.
      Points A and B are points where two separate tangents intersect the circle. $$d$$° = °.

      An image in a quiz
      Correct Answer: 35, 35°, 35 degrees

      Q5.
      The tangent shown intersects the circle at point T. $$x$$° = °.

      An image in a quiz
      Correct Answer: 48, 48°, 48 degrees

      Q6.
      Point T is a point on the circumference where a tangent intersects the circle. $$y$$° = °.

      An image in a quiz
      Correct Answer: 77, 77°, 77 degrees

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