Coordinates
Section: Coordinate geometry | Syllabus: Cambridge IGCSE Mathematics (0580)
Reading and Plotting Coordinates
Coordinates describe the exact position of a point on a grid, using two numbers written as an ordered pair.
- A coordinate is written as (x, y) - the x-coordinate (horizontal position) is always given first, the y-coordinate (vertical position) second
- The origin, where the x-axis and y-axis cross, has coordinates (0, 0)
- The plane is divided into four quadrants by the two axes, numbered 1st to 4th anticlockwise starting from the top-right
- The signs of the coordinates show which quadrant a point lies in: (+,+) 1st quadrant, (−,+) 2nd, (−,−) 3rd, (+,−) 4th
Worked Example: Identifying a Quadrant from Coordinates
- Question: State which quadrant the point (-5, 3) lies in.
- Step 1: The x-coordinate is negative and the y-coordinate is positive
- Step 2: This sign pattern (−,+) matches the 2nd quadrant
- Answer: 2nd quadrant
Common Mistakes
MistakeWriting the y-coordinate before the x-coordinate, e.g. reading (4,-2) as "x=-2, y=4"
Fixcoordinates are always written in the order (x, y) - the horizontal value always comes first
MistakeMiscounting which quadrant is which, especially mixing up the 2nd and 4th
Fixquadrants are numbered anticlockwise starting from the top-right (1st) - sketch a quick diagram if unsure
Finding Coordinates to Complete a Shape
In many problems, three vertices of a shape are given and the fourth must be found using the shape's known properties, such as being a rectangle or a parallelogram.
- For a parallelogram (or rectangle), opposite sides are equal and parallel - moving from one vertex to an adjacent one gives the same horizontal and vertical shift as the opposite side
- Sketching the shape roughly on a grid, even without exact scale, makes it much easier to see which vertex is missing and where it should go
- Check the answer by confirming that both pairs of opposite sides have the same shift in x and in y
Worked Example: Finding a Missing Vertex
- Question: ABCD is a parallelogram. A = (1, 2), B = (5, 3), C = (6, 6). Find the coordinates of D.
- Step 1: Find the shift from A to B: (5-1,\ 3-2)=(4,1)
- Step 2: Since ABCD is a parallelogram, side DC has the same shift as side AB, so going from C back to D is the reverse shift, (-4,-1)
- Step 3: D=(6-4,\ 6-1)=(2,5)
- Answer: D = (2, 5)
Common Mistakes
MistakeApplying the shift from the wrong pair of vertices, mixing up which sides are opposite each other
Fixlabel the vertices in order around the shape (A, B, C, D) - then AB is opposite DC, and BC is opposite AD
MistakeAdding the shift in the wrong direction, e.g. adding instead of subtracting
Fixsketch the shape roughly first - a quick sketch shows immediately whether the missing vertex should be shifted forward or backward from a known point
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