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9.1 Coordinates

GCSE Maths Foundation · Edexcel · Chapter 9: Graphs

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: reading and plotting points in all four quadrants, and finding midpoints. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Read and plot coordinates in the first quadrant and in all four quadrants
  • Use coordinates to find missing corners of shapes
  • Find the midpoint of a line segment from a diagram or by calculation
  • Work backwards from a midpoint to find an end point

Key words

  • Coordinates: A pair of numbers (x, y) that gives the position of a point. x first, then y.
  • Origin: The point (0, 0), where the x-axis and the y-axis cross.
  • x-axis, y-axis: The x-axis goes across (horizontal). The y-axis goes up and down (vertical).
  • Quadrant: One of the four regions made by the two axes.
  • Line segment: Part of a line with two end points.
  • Midpoint: The point exactly halfway along a line segment.

Reading and plotting coordinates

Coordinates are always written (x, y). Start at the origin. The x-coordinate tells you how far across to go. The y-coordinate tells you how far up to go. A good way to remember: “along the corridor, then up the stairs”.

Key fact: Mark a point with a small cross (×) or a dot, and label it with its letter.

Coordinates in four quadrants

The axes can be extended past 0. Negative x means go left. Negative y means go down. The two axes split the grid into four quadrants.

Worked example: plotting with negatives

Plot Q(−4, 1): go 4 left, then 1 up.
Plot R(−2, −2): go 2 left, then 2 down.
Plot S(1, −3): go 1 right, then 3 down.

Answer: see Q, R and S on the grid

  • (5, 0); Where it is on the x-axis (y = 0)
  • (0, −3); Where it is on the y-axis (x = 0)
  • (−2, 6); Where it is second quadrant: left of the y-axis, above the x-axis
  • (4, −1); Where it is fourth quadrant: right of the y-axis, below the x-axis

Coordinates and shapes

You can use what you know about a shape to find a missing corner (vertex). Sketch the points first. Then use the facts, for example “a rectangle has opposite sides equal and parallel”.

Worked example: a missing vertex

A(−2, 2), B(4, 2) and C(4, −2) are three corners of a rectangle ABCD. Find D.
D must be directly below A (same x-coordinate, −2) and level with C (same y-coordinate, −2).

Answer: D is (−2, −2)

Worked example: side lengths from coordinates

AB goes from x = −2 to x = 4, so AB = 4 − (−2) = 6 units.
BC goes from y = 2 to y = −2, so BC = 2 − (−2) = 4 units. The area is 6 × 4 = 24 square units.

Answer: 6 units by 4 units, area 24 square units

The midpoint of a line segment

The midpoint is halfway between the two end points. Its x-coordinate is halfway between the two x-coordinates, and its y-coordinate is halfway between the two y-coordinates.

Key fact: Midpoint of (x1, y1) and (x2, y2) = ( (x1 + x2) ÷ 2 , (y1 + y2) ÷ 2 ). Add, then halve.

Worked example: by calculation

Find the midpoint of A(1, 5) and B(7, 1).
x: (1 + 7) ÷ 2 = 8 ÷ 2 = 4. y: (5 + 1) ÷ 2 = 6 ÷ 2 = 3.

Answer: M(4, 3)

Worked example: with negative numbers

Find the midpoint of P(−3, 4) and Q(5, −2).
x: (−3 + 5) ÷ 2 = 2 ÷ 2 = 1. y: (4 + (−2)) ÷ 2 = 2 ÷ 2 = 1.

Answer: (1, 1)

Worked example: the centre of a circle

The points (−1, 6) and (7, 2) are the ends of a diameter of a circle. Find the centre.
The centre is the midpoint of the diameter: x = (−1 + 7) ÷ 2 = 3, y = (6 + 2) ÷ 2 = 4.

Answer: centre (3, 4)

Working backwards. If you know one end and the midpoint, the other end is the same step again. Find the step from the end to the midpoint, then add it on once more.

Worked example: finding an end point

M(4, 3) is the midpoint of AB. A is (1, 1). Find B.
From A to M: x goes up by 3 (1 to 4), y goes up by 2 (1 to 3).
Do the same again from M: x = 4 + 3 = 7, y = 3 + 2 = 5.

Answer: B is (7, 5)

Common mistakes

  • Writing the coordinates the wrong way round. It is always (x, y): across first, then up or down.
  • Going the wrong way for a negative number. Negative x is to the left; negative y is down.
  • Subtracting the coordinates for a midpoint. You add them, then halve.
  • Sign slips such as (−3 + 5) ÷ 2 = −4. Use a number line if you are unsure: −3 + 5 = 2.
  • Thinking (0, 4) is on the x-axis. It has x = 0, so it is on the y-axis.

Revision questions

Try these, then check your answers below each one.

Question 1. Point P is 3 units to the left of the origin and 5 units up. Write down the coordinates of P.

Answer: (-3,5)

Working: Left 3 gives x = -3; up 5 gives y = 5. P is (-3, 5).

Question 2. Find the midpoint of the line segment joining (−4, 1) and (6, −5).

Answer: (1,-2)

Working: x: (-4 + 6) ÷ 2 = 1. y: (1 + (-5)) ÷ 2 = -2. Midpoint (1, -2).

Question 3. Sam says, “The point (0, −3) lies on the x-axis.” Is Sam correct?

Answer: No

Working: Points on the x-axis have y = 0. (0, −3) has x = 0, so it lies on the y-axis. Sam is wrong.

Related lessons

Continue in the full course

This lesson is section 9.1 (EGCMF0901) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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