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      Advanced problem solving with linear graphs

      Lesson details

      Learning outcome

      I can use my knowledge of linear graphs to solve problems.

      Key learning points

      1. Knowledge of the geometric properties of shapes can be used here.
      2. Knowledge of ratio can be used here.

      Keywords

      • Tangent (of a circle) - The tangent (of a circle) is a line that intersects the circle exactly once.

      Common misconception

      P is a point such that AP : PB is in the ratio of 1 : 3 means that point P is a third of the distance of AB from A.

      There are 4 equal parts in total, meaning that point P is a quarter of the distance of AB from A.

      Teacher tip

      The feedback slides show where annotations have been added to the given diagram in order to support answering the question. Encourage pupils to annotate any given diagrams with the information they deduce in order to support reaching a solution.

      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.
      Two lines are perpendicular if their gradients are the negative __________ of one another.

      gradient
      multiplier
      product
      Correct answer: reciprocal

      Q2.
      Match each line with a line that is perpendicular to it.

      Correct Answer:$$y=3x-8$$,$$y=-{1\over3}x-11$$

      $$y=-{1\over3}x-11$$

      Correct Answer:$$y={1\over2}x+8$$,$$y=10-2x$$

      $$y=10-2x$$

      Correct Answer:$$y=7-{2\over3}x$$,$$y={3\over2}x-3$$

      $$y={3\over2}x-3$$

      Correct Answer:$$y={3\over2}x-10$$,$$y=-{2\over3}x+8$$

      $$y=-{2\over3}x+8$$

      Q3.
      What is the equation of this line?

      An image in a quiz
      $$y=1+3x$$
      $$y=1+{1\over3}x$$
      $$y={1\over3}x-1$$
      $$y=-{1\over3}x-1$$
      Correct answer: $$y=1-{1\over3}x$$

      Q4.
      What is the equation of the straight line that goes through the coordinates (-5, -23) and (3, 17)?

      $$y=2-5x$$
      $$y=2x+5$$
      Correct answer: $$y=5x+2$$
      $$y=5x-2$$
      $$y=2x-5$$

      Q5.
      Find the coordinates of the point P when AP : PB is in the ratio 1 : 5.

      An image in a quiz
      (3, 2)
      (2, 3)
      (4 3)
      Correct answer: (3, 5)
      (1, 5)

      Q6.
      What is the coordinate of C when AB : BC is in the ratio 1 : 2?

      An image in a quiz
      (0.5, 1)
      (5, -8)
      (7, -12)
      Correct answer: (11, -20)
      (15, -28)

      6 Questions

      Q1.
      In a circle, a tangent and a __________ meet at a right angle.

      chord
      circumference
      Correct answer: radius

      Q2.
      The grid shows a circle, its radius and a tangent which touches the circumference at (3, -4). What is the equation of the tangent line?

      An image in a quiz
      $$y=2x+5$$
      Correct answer: $$y=2x-10$$
      $$y=2x-4$$
      $$y={1\over2}x+5$$
      $$y={1\over2}x-10$$

      Q3.
      The grid shows a circle, its radius and a tangent which touches the circumference at (3,-4). The radius is a line segment of the line ...

      An image in a quiz
      $$y=-5-2x$$
      $$y=-2x-2.5$$
      $$y=-{1\over2}x-5$$
      $$y=-{1\over2}x+{5\over2}$$
      Correct answer: $$y=-{1\over2}x-{5\over2}$$

      Q4.
      In this graph the radius is a line segment of the line $$y=-{1\over2}x-{5\over2}$$ and the tangent is line $$y=2x-10$$. How can you tell they meet at a right angle?

      An image in a quiz
      $$(-4)=-{1\over2}(3)-{5\over2}$$
      $$(-4)=2(3)-10$$
      $$-{5\over2}\times{10}=-25$$
      Correct answer: $$-{1\over2}\times{2}=-1$$
      $$-{1\over2}\times{2}=1$$

      Q5.
      Point P is such that AP : PB is in the ratio 1 : 2. Find the area of triangle OPB.

      An image in a quiz
      18 sq units
      27 sq units
      Correct answer: 36 sq units
      54 sq units

      Q6.
      Point P is such that AP : PB is in the ratio 5 : 1. The area of triangle OAP is square units.

      An image in a quiz
      Correct Answer: 45

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