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Coordinate Geometry

Coordinate Geometry and Graphs

Pythagoras on the grid

  • The distance between two points is the hypotenuse of a right-angled triangle drawn with one horizontal and one vertical side
  • The horizontal side is the difference in the x values and the vertical side is the difference in the y values
  • So the length is √((x₂ − x₁)² + (y₂ − y₁)²)
Two points (x₁, y₁) and (x₂, y₂) joined by a sloping line, with a horizontal side b and a vertical side c completing a right-angled triangle whose hypotenuse a is the distance between them, so a² = b² + c²
Source: Length of a line by Save My Exams
  • Squaring removes any minus sign, so it does not matter which point is taken first
  • Like the midpoint formula, this one is not printed in the paper
  • Writing the two differences down before squaring them earns a mark on its own, and the completed substitution earns another
  • Put brackets around negative coordinates as you subtract them, since a subtraction of a negative is where the sign errors happen
Worked example

A length that is a whole number

C is the point (2, −5) and D is the point (14, 4). Calculate the length of CD.

Solution:

  • Difference in x: 14 − 2 = 12
  • Difference in y: 4 − (−5) = 9
  • Square and add: 12² + 9² = 144 + 81 = 225
  • Take the square root: √225 = 15
  • The length of CD is 15 units

Leaving the answer as a surd

  • Where the number under the root is not a perfect square, the exact answer is a surd
  • Simplify it by taking out the largest square factor, exactly as in the surds topic
  • Questions on this specification regularly ask for the length in surd form, or state that it equals k√n and ask for k
  • A decimal is only acceptable when the question asks for one, and then the required accuracy is stated
Exam tip

The List of formulas on page 2 does not carry the midpoint or the distance between two points, so both must be memorised. Write the two differences, or the two averages, on a line before combining them: each midpoint coordinate is credited separately, and for a length the differences alone earn a mark and the full substitution another. Mark schemes accept an exact surd as a full-mark answer.

Worked example

A length in surd form

A is the point (−1, 4) and B is the point (5, 1). Find the length of AB, giving your answer as a surd in its simplest form.

Solution:

  • Difference in x: 5 − (−1) = 6
  • Difference in y: 1 − 4 = −3
  • Square and add: 6² + (−3)² = 36 + 9 = 45
  • So the length is √45
  • The largest square factor of 45 is 9, and 45 = 9 × 5
  • √45 = √9 × √5 = 3√5
  • The length of AB is 3√5 units