Coordinate Geometry with Straight Lines and Circles

Coordinate Geometry with Straight Lines and Circles Lessons

4 Part Lesson
Circles and Three Points
4 Part Lesson
Circles and Chords
4 Part Lesson
Equation of a Circle
4 Part Lesson
Intersections with Circles
4 Part Lesson
Parallel and Perpendicular Lines
4 Part Lesson
Equation of a Line Between Two Points
4 Part Lesson
Gradient and Midpoint of Line Segments

Prerequisite Knowledge

  • Find the gradient of a straight-line graph.
  • Draw straight line graphs.
  • Rearrange straight line equations into the form y = mx + c
  • Find lengths and midpoints between two coordinate pairs.
  • Complete the square of a quadratic function.
  • Know and be able to apply Pythagoras’ Theorem
  • Construct a circle in the form x2 + y2 = r2.
  • Solve quadratic and linear equations simultaneously using the method of substitution
  • A tangent is perpendicular to the radius of the circle at the point of intersection.

Success Criteria

  • Understand and use the equation of a straight line, including the forms y = mx + c and m(x – x1) = y – y1
  • Gradient conditions for two straight lines to be parallel or perpendicular
  • Be able to use straight line models in a variety of contexts
  • Understand and use the coordinate geometry of the circle including using the equation of a circle in the form (x – a)2 + (y – b)2 = r2
  • Completing the square to find the centre and radius of a circle.
  • Use of the following properties:
    • the angle in a semicircle is a right angle
    • the perpendicular from the centre to a chord bisects the chord
    • the radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point

Teaching Points

  • Points are collinear if they all lie on the same straight line.
  • The equation of a straight line can be given as y = mx + c or ax + by + c = 0. Students need to be able to convert between the two forms.
  • The equation a line in the form m(x – x1) = y – y1 can be derived from the gradient of a straight line where m = (y -y1) / (x – x1)
  • Parallel lines have the same gradient. Perpendicular lines have a negative reciprocal gradient.
  • If two lines are perpendicular, the product of their gradients is -1.
  • The equation of a circle with centre (a, b) can be given in the forms (x – a)2 + (y – b)2 = r2 and x2 + y2 + 2fx + 2gy + c = 0.

Misconceptions

  • Students can often become confused with the algebraic workings due to not drawing diagrams or diagrams lacking sufficient detail.
  • A common mistake is to write the correct gradient of line in the form y = mx + c but write it incorrectly when converting the equation into the form ax + by + c = 0.
  • The coordinates of the centre of a circle are sometimes given with the negatives.  For instance, (x – 3)2 + (y – 4)2 = r2 can have the incorrect centre as (-3, -4).
  • Students can often become confused with the algebraic workings due to not drawing diagrams or diagrams lacking sufficient detail.

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