Coordinates in Four Quadrants
This topic covers the Cartesian coordinate system, which uses x and y values to locate points in a 2D plane. Mastering this is essential as it forms the basis for all graphing, geometry, and vector questions in the ESAT.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The plane is defined by a horizontal x-axis and a vertical y-axis, which intersect at the origin (0,0).
- A point's location is given by coordinates (x, y), where x is the horizontal distance from the origin and y is the vertical distance.
- The axes divide the plane into four quadrants, numbered anti-clockwise from the top right.
- Quadrant I: x is positive (+), y is positive (+).
- Quadrant II: x is negative (-), y is positive (+).
- Quadrant III: x is negative (-), y is negative (-).
- Quadrant IV: x is positive (+), y is negative (-).
- The horizontal distance between two points with the same y-value is the absolute difference of their x-coordinates. The vertical distance between two points with the same x-value is the absolute difference of their y-coordinates.
Diagram
Formulae
dhorizontal = |x2 - x1| To find the distance between two points that lie on the same horizontal line, e.g., (x1, y) and (x2, y).
dvertical = |y2 - y1| To find the distance between two points that lie on the same vertical line, e.g., (x, y1) and (x, y2).
Definitions
- Cartesian Coordinates
- An ordered pair of numbers (x, y) that specifies the position of a point on a plane relative to the origin.
- Origin
- The point (0,0) where the x-axis and y-axis intersect.
- Quadrant
- One of the four regions into which the x and y axes divide the coordinate plane.
Worked example
A rectangle ABCD has vertices A at (-5, 6) and B at (2, 6). The length of side BC is 10 units. If vertex C is in the fourth quadrant, find the coordinates of vertices C and D.
- 1
First, find the length of side AB.
Since the y-coordinates are the same, this is a horizontal line.
The length is the difference in x-coordinates:
2 - (-5) = 7 units - 2
Side BC is perpendicular to AB and has a length of 10 units.
Therefore, BC must be a vertical line segment.
- 3
To find the coordinates of C, we start at B(2, 6) and move vertically.
Since C is in the fourth quadrant, its y-coordinate must be negative.
We move down 10 units from B:
y = 6 - 10 = -4The x-coordinate remains the same.
So, C is at (2, -4).
- 4
To find D, we complete the rectangle.
D must have the same x-coordinate as A (-5) and the same y-coordinate as C (-4).
So, D is at (-5, -4).
- 5
Check:
Point D(-5, -4) is in the third quadrant, which is consistent with the shape of the rectangle.
Answer: C = (2, -4), D = (-5, -4)
Common mistakes
- ×Swapping the x and y coordinates. Remember the mnemonic: 'walk along the corridor (x-axis), then climb the stairs (y-axis)'.
- ×Making sign errors when calculating distances between points with negative coordinates. For example, the distance from x=-5 to x=2 is 2 - (-5) = 7, not 2-5 = -3.
- ×Confusing the quadrant numbering. Quadrants are numbered I, II, III, IV in an anti-clockwise direction starting from the top-right.
No-calculator tips
- ✓Always sketch a quick, rough diagram of the axes and points. This visual aid makes it much easier to see relationships and avoid simple sign errors.
- ✓To find the distance between two points on an axis (e.g., -5 and 2), treat it as a journey. To get from -5 to 0 is 5 steps, then from 0 to 2 is 2 steps. The total distance is 5 + 2 = 7.