Orienting Yourself: The Use of Coordinates · Lesson 3 of 5
The 2-D Cartesian Coordinate System
“At last, x and y agree to meet—strictly at right angles.”
• Understand the x-axis, y-axis and origin. • Interpret positive and negative directions on both axes. • Locate points on the axes and in all four quadrants. • Read and write ordered pairs in the correct order. • Explain the geometric meaning of the x-coordinate and y-coordinate.
From One Dimension to Two
The room grid helps us understand the basic idea of position: an object can be located by describing how far it is from fixed reference lines. Mathematics takes this idea and gives it a standard form called the Cartesian coordinate system.
The Cartesian coordinate system allows us to describe the exact position of any point on a flat surface using numbers. Instead of saying that a point is “somewhere to the right” or “a little above”, we can give its precise location using a pair of coordinates. Because the same system can be used on maps, graphs, diagrams, and geometric figures, it becomes a common mathematical language for studying positions and relationships in two dimensions.
Think first about a number line. It has only one direction of movement: left or right. Because of this, only one number is needed to locate a point. For example, the number 4 tells us that the point lies four units to the right of zero, while 3 tells us that it lies three units to the left.
A flat surface, however, is different. A point can move not only left or right, but also up or down. This means that one number is no longer enough to describe its position completely.
To solve this problem, we use two number lines placed perpendicular to each other. The horizontal number line tells us how far a point is to the left or right, while the vertical number line tells us how far it is above or below.
Together, these two perpendicular number lines form the coordinate plane. Every point on this plane can be located uniquely using an ordered pair of numbers, such as (3, 2) or (-4, 5). The first number describes the horizontal position, and the second describes the vertical position.
This simple idea allows us to convert geometric positions into numerical information, making it possible to measure distances, study shapes, draw graphs, and solve many problems using both algebra and geometry.
The horizontal coordinate axis is called the x-axis.
The vertical coordinate axis is called the y-axis.
The point where the x-axis and y-axis intersect is called the origin. Its coordinates are (0, 0).
Positive and Negative Directions
Starting from the origin, distances to the right on the x-axis are positive and distances to the left are negative. On the y-axis, distances upward are positive and distances downward are negative. These signs are not decorations; they tell us direction as well as distance.
| Movement from O | Sign |
|---|---|
| Right along x-axis | + |
| Left along x-axis | − |
| Up along y-axis | + |
| Down along y-axis | − |
Points on the Axes
A point on the x-axis has no vertical displacement from the x-axis, so its y-coordinate is zero. A point on the y-axis has no horizontal displacement from the y-axis, so its x-coordinate is zero.
Problem
Describe B = (4.5, 0), G = (0, −4.5) and H = (0, 4).
- 1.B has y-coordinate 0, so it lies on the x-axis. Since x = 4.5 is positive, it lies 4.5 units to the right of O.
- 2.G has x-coordinate 0, so it lies on the y-axis. Since y = −4.5, it lies 4.5 units below O.
- 3.H also lies on the y-axis. Since y = 4, it lies 4 units above O.
The Cartesian Plane and Its Quadrants
The plane containing the x-axis and y-axis is called the Cartesian plane, coordinate plane or xy-plane.
The two coordinate axes divide the plane into four regions called quadrants.
The quadrants are numbered counter-clockwise, starting from the upper-right region. The signs of x and y immediately tell us which quadrant contains a point.
| Quadrant | Sign of x | Sign of y | Coordinate pattern |
|---|---|---|---|
| I | + | + | (+, +) |
| II | − | + | (−, +) |
| III | − | − | (−, −) |
| IV | + | − | (+, −) |
What Does (x, y) Actually Mean?
For a point P(x, y), the first coordinate x describes its horizontal position. Geometrically, it is the signed perpendicular distance from the y-axis, measured along the x-direction. The second coordinate y describes its vertical position; it is the signed perpendicular distance from the x-axis.
The first number in (x, y). It gives the signed horizontal position of the point, or its signed distance from the y-axis.
The second number in (x, y). It gives the signed vertical position of the point, or its signed distance from the x-axis.
The Order of Coordinates Matters
Coordinates form an ordered pair. The first number always belongs to the x-direction and the second to the y-direction. Swapping them usually changes the point.
Problem
Compare S(3, −5) and Q(−5, 3).
- 1.S has x = 3 and y = −5, so it lies in Quadrant IV.
- 2.Q has x = −5 and y = 3, so it lies in Quadrant II.
- 3.The same numbers appear, but their order and signs differ.
- 4.Therefore the points lie in different parts of the plane.
Reading Coordinates Without Plotting
| Point | Reasoning | Location |
|---|---|---|
| (6, 2) | Both coordinates positive | Quadrant I |
| (−4, 7) | x negative, y positive | Quadrant II |
| (−3, −8) | Both negative | Quadrant III |
| (5, −6) | x positive, y negative | Quadrant IV |
| (0, 9) | x = 0 | y-axis |
| (−7, 0) | y = 0 | x-axis |
| (0, 0) | Both zero | Origin |
Parallel Lines and Shared Coordinates
Two points with the same y-coordinate lie on a horizontal line, parallel to the x-axis. Two points with the same x-coordinate lie on a vertical line, parallel to the y-axis. This is extremely useful when reading room layouts, doors and rectangles.
Problem
Without plotting, compare A(−5, 2), B(−5, −3) and C(4, 2).
- 1.A and B share x = −5, so AB is vertical and parallel to the y-axis.
- 2.A and C share y = 2, so AC is horizontal and parallel to the x-axis.
- 3.A vertical line and a horizontal line meet at a right angle, so AB is perpendicular to AC.
Practice Problems
- State the quadrant or axis containing: (4,7), (−6,2), (−3,−5), (8,−1), (0,6), (−9,0).
- Point W has x-coordinate −5. A point H lies on the vertical line through W. What must the x-coordinate of H be? Which quadrants could H lie in?
- For R(3,0), A(0,−2), M(−5,−2) and P(−5,2), identify one horizontal side, one vertical side and a pair of perpendicular sides without plotting first.
- Explain why (x,y) and (y,x) are usually different points.
- What would be impossible to represent if a coordinate system allowed only non-negative coordinates?
Key Takeaways
• The x-axis is horizontal, the y-axis is vertical, and they meet at O(0,0). • Right and up are positive; left and down are negative. • Points on the x-axis have form (x,0); points on the y-axis have form (0,y). • The axes divide the plane into four quadrants with fixed sign patterns. • Coordinates are ordered: x comes first, y comes second.
Next, we use coordinate differences and the Baudhāyana–Pythagoras theorem to find the distance between any two points in the plane.