Coordinate Geometry is the magical bridge between algebra and geometry: it lets you pin down the exact position of any point on a flat plane using just two numbers. No more vague 'somewhere over there' → every point gets a precise address!
The Plane
Two number lines crossing at right angles create the Cartesian plane.
Axes
The horizontal x-axis and vertical y-axis meet at the origin O.
Coordinates
Every point has an ordered pair (x, y) — its unique address.
Quadrants
The axes split the plane into four regions numbered I, II, III, IV.
1. Why do we need Coordinate Geometry?
Imagine trying to describe exactly where a chair is in a room. You might say 'it is 3 metres from the left wall and 2 metres from the front wall.' Those two distances together fix the chair's position perfectly. Coordinate Geometry does exactly this for points on a flat surface (a plane). The brilliant French mathematician René Descartes invented this system in the 17th century — the legend says he watched a fly on his ceiling and wondered how to describe its position using numbers. Because of him, the system is called the Cartesian system. It turns shapes and positions into numbers, so we can use algebra (equations) to solve geometry problems.
2. The Cartesian Plane: axes and origin
Take a horizontal number line and call it the x-axis (also written X′X). Take a vertical number line and call it the y-axis (Y′Y). Place them so they cross each other at right angles (90°). The point where they cross is called the origin, written as O, and it has coordinates (0, 0). These two axes together are called the coordinate axes, and the flat surface they sit on is the Cartesian plane or coordinate plane. On the x-axis, numbers to the right of O are positive and numbers to the left are negative. On the y-axis, numbers above O are positive and numbers below O are negative.
3. Coordinates of a point: abscissa and ordinate
To locate any point P, we measure two distances. The x-coordinate is the distance of the point from the y-axis, measured along the x-direction. It is also called the abscissa. The y-coordinate is the distance of the point from the x-axis, measured along the y-direction. It is also called the ordinate. We always write the coordinates together as an ordered pair (x, y), with the x-coordinate first and the y-coordinate second, separated by a comma and enclosed in brackets. The word 'ordered' is important: (3, 5) is a completely different point from (5, 3). Order matters!
4. The four quadrants
The two axes divide the plane into four parts called quadrants, numbered anticlockwise starting from the top-right. Quadrant I (top right): x is +, y is + → (+, +). Quadrant II (top left): x is −, y is + → (−, +). Quadrant III (bottom left): x is −, y is − → (−, −). Quadrant IV (bottom right): x is +, y is − → (+, −). Knowing the signs immediately tells you which quadrant a point lives in, and vice versa. For example, the point (−4, 7) has a negative x and positive y, so it sits in Quadrant II.
5. Points lying ON the axes
What about points that sit right on an axis? A point on the x-axis has its y-coordinate equal to 0, so it looks like (x, 0) — for example (6, 0) or (−2, 0). A point on the y-axis has its x-coordinate equal to 0, so it looks like (0, y) — for example (0, 4) or (0, −3). Points on the axes do not belong to any quadrant; they are on the boundary. And the origin (0, 0) lies on both axes.
6. Plotting a point in the plane
To plot a point, say A(3, 2): start at the origin O. Move 3 units to the right along the x-axis (because x = +3). From there, move 2 units up parallel to the y-axis (because y = +2). Mark the point and label it A. Always read the x-coordinate first (left/right) and then the y-coordinate (up/down). For a negative coordinate you simply move in the opposite direction. To plot B(−2, −3): move 2 units left, then 3 units down.
7. Reading coordinates from a graph
The reverse skill is just as useful. Given a marked point on a graph, drop a perpendicular (a straight line at 90°) from the point onto the x-axis to read its x-coordinate, and another perpendicular onto the y-axis to read its y-coordinate. Combine them as the ordered pair (x, y). Choosing a sensible scale (for example, 1 small square = 1 unit) makes plotting and reading accurate, and the scale on both axes does not have to be the same.
- A point is written as an ordered pair (x, y) — x first, y second.
- x-coordinate = abscissa = distance from the y-axis.
- y-coordinate = ordinate = distance from the x-axis.
- Origin O = (0, 0); it lies on both axes.
- Point on x-axis → (x, 0); point on y-axis → (0, y).
- Quadrant signs: I (+, +), II (−, +), III (−, −), IV (+, −).
- Quadrants are numbered anticlockwise from the top-right.
In which quadrant or on which axis do each of the following points lie? A(−3, 5), B(7, 0), C(−2, −6), D(0, −4).
- For A(−3, 5): x is negative, y is positive → signs (−, +) → this matches Quadrant II.
- For B(7, 0): the y-coordinate is 0, so the point lies ON the x-axis (not in any quadrant).
- For C(−2, −6): both coordinates are negative → signs (−, −) → this matches Quadrant III.
- For D(0, −4): the x-coordinate is 0, so the point lies ON the y-axis (below the origin).
Plot the points P(2, 3) and Q(3, 2) on the same plane. Are they the same point? Also state the coordinates of the point that is 4 units to the left of the origin on the x-axis.
- To plot P(2, 3): from origin O, move 2 units right along the x-axis, then 3 units up. Mark P.
- To plot Q(3, 2): from origin O, move 3 units right along the x-axis, then 2 units up. Mark Q.
- Compare: P is higher and slightly left of Q, while Q is lower and slightly right. They occupy different positions.
- This shows order matters: (2, 3) is NOT the same as (3, 2). They are two distinct points, both in Quadrant I.
- A point 4 units left of the origin on the x-axis has x = −4 and y = 0, so it is (−4, 0).
'x comes before y in the alphabet, so x comes first in the bracket.' And for the quadrant order, picture writing the letter C backwards — you start top-right (I), sweep to top-left (II), down to bottom-left (III), and finish bottom-right (IV): anticlockwise! For axes: 'Run before you climb' — first run sideways (x), then climb up/down (y).
The most common mistake is swapping the coordinates — plotting (5, 2) where (2, 5) should go. Always remember: the FIRST number is the horizontal move (x, left/right) and the SECOND number is the vertical move (y, up/down). The second common slip is saying a point on an axis is 'in a quadrant' — points on the axes belong to NO quadrant.
Q1. Write the name of the point whose ordinate is 0 and abscissa is 5. In which part of the plane does it lie?
Answer: Ordinate is the y-coordinate, so y = 0; abscissa is the x-coordinate, so x = 5. The point is (5, 0). Since the y-coordinate is 0, the point lies ON the x-axis (to the right of the origin), so it is not in any quadrant.
Q2. The points (−5, 2), (−5, −2) and (−5, 7) all have the same x-coordinate. What can you say about their positions, and in which quadrants do they lie?
Answer: All three points have x = −5, meaning each is exactly 5 units to the LEFT of the y-axis. So they all lie on a single vertical line parallel to the y-axis. Checking signs: (−5, 2) has (−, +) → Quadrant II; (−5, 7) has (−, +) → Quadrant II; (−5, −2) has (−, −) → Quadrant III. So two are in Quadrant II and one is in Quadrant III, but all lie on the same vertical line x = −5.
Q3. Plot the points A(1, 1), B(4, 1) and C(4, 4) and D(1, 4). What figure is formed when you join them in order?
Answer: Plot each point: A is 1 right and 1 up; B is 4 right and 1 up; C is 4 right and 4 up; D is 1 right and 4 up. AB is horizontal with length 4 − 1 = 3 units. BC is vertical with length 4 − 1 = 3 units. CD is horizontal with length 3 units. DA is vertical with length 3 units. All four sides equal 3 units and all angles are 90°, so ABCD is a square of side 3 units. (Its area = 3 × 3 = 9 square units.)
Q4. If the x-coordinate of a point is positive and its y-coordinate is negative, in which quadrant does it lie? Give one example and explain how you would plot it.
Answer: The signs are (+, −), which matches Quadrant IV (bottom-right region). An example is the point (5, −3). To plot it: start at the origin O, move 5 units to the RIGHT along the x-axis (because x = +5), then from there move 3 units DOWN parallel to the y-axis (because y = −3). Mark and label the point. It will sit below the x-axis on the right side, confirming Quadrant IV.
- ✅ The Cartesian plane is made of the x-axis and y-axis crossing at the origin (0, 0).
- ✅ Every point has a unique ordered pair (x, y): x = abscissa (from y-axis), y = ordinate (from x-axis).
- ✅ Four quadrants, numbered anticlockwise: I (+,+), II (−,+), III (−,−), IV (+,−).
- ✅ Points on the axes have a 0 coordinate and belong to no quadrant; plot by 'run then climb'.
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