Orienting Yourself: The Use of Coordinates
Why This Matters
Imagine you order food online. The delivery person finds your exact house out of lakhs of houses in the city. How? Your address is really a set of directions: which area, which lane, which house number. Each piece narrows things down until only one spot is left.
Maths needs the same idea. We often want to point at one exact spot on a flat surface — a page, a screen, a map — and say “here, this one, no other”. A single number is not enough. On a phone screen, “go 5 across” still leaves a whole vertical line of pixels to choose from. You need a second number to pin it down: “5 across and 3 up”. Two numbers, one exact point.
That simple idea — using a pair of numbers to fix a location — is one of the most powerful tools in all of mathematics. It is called the coordinate system, and it is everywhere. It is how Google Maps knows where you are. It is how a video game draws a character on the screen. It is how a graph shows a cricket score climbing over the overs.
This idea is also very old, with deep roots in India. Thousands of years ago, the cities of the Sindhu-Sarasvatī Civilisation were laid out on a precise grid, with streets running North–South and East–West about 10 metres apart. A merchant could find any shop by counting blocks in two directions from the centre. That is a coordinate system in real life. Much later, Brahmagupta (around 628 CE) worked out the rules for zero and negative numbers — and without those, the four-part coordinate plane you will study here could not exist.
In this chapter you will learn to do three things on your own: name any point in a plane with a pair of numbers, plot a point that someone gives you, and measure the straight-line distance between any two points.
The Big Idea
Here is the whole chapter in one sentence.
The Big Idea: Place two number lines so they cross at right angles — one going across, one going up. Then every point on the flat surface gets a unique name: a pair of numbers (x, y), where x says how far across it is and y says how far up. One pair of numbers, one exact point — and the same pair never points to two different places.
Everything else — the axes, the origin, the four quadrants, plotting, reading coordinates, measuring distance — is just working out the details of this one idea.
From One Number Line to Two
You already met the number line in earlier classes. Let us refresh it, because the coordinate system is built directly on top of it.
This chapter stands on the number line from earlier grades, so here is a quick reminder before we extend it.
Figure 1.1 below shows a number line, with the point “3” marked on it.
But a flat surface is two-dimensional (2-D for short). It has width and height. So one number is no longer enough. To fix a point on a page you need to know two things: how far across, and how far up or down.
The clever fix is simple. We use two number lines instead of one, and we cross them at right angles:
- One number line is laid horizontal (across). We call it the x-axis.
- The other number line is laid vertical (up and down). We call it the y-axis.
The point where the two lines cross is called the origin, written O. It is the “starting corner” — the 0 of both number lines at once. Its coordinates are (0, 0).
Together the two crossed lines are called the coordinate axes (“axes” is just the plural of “axis”). Figure 1.2 shows this structure.
Look carefully at the four points in Figure 1.2 — they teach the sign rules.
- B = (4.5, 0) sits on the x-axis, 4.5 units to the right of O. Right is positive, so its x-value is +4.5. It is not above or below the axis, so its y-value is 0.
- E = (−2.9, 0) also sits on the x-axis, but to the left of O. Left is negative, so its x-value is −2.9.
- H = (0, 4) sits on the y-axis, 4 units above O. Up is positive, so its y-value is +4. It is not left or right, so its x-value is 0.
- G = (0, −4.5) sits on the y-axis, 4.5 units below O. Down is negative, so its y-value is −4.5.
So there is a neat pattern. A point on the x-axis always looks like (x, 0) — its second number is 0. A point on the y-axis always looks like (0, y) — its first number is 0. A 0 in the pair is a clue that the point sits exactly on an axis.
A point sits on the y-axis, 7 units below the origin. What are its coordinates?
What an Ordered Pair Really Means
Now to the heart of it. The coordinates of any point P are written as a pair (x, y). This is called an ordered pair — “ordered” because the order matters a lot. Here is exactly what each number means:
- x (the x-coordinate) = how far the point is from the y-axis, measured across. It tells you how far left or right you are.
- y (the y-coordinate) = how far the point is from the x-axis, measured up or down. It tells you how far up or down you are.
The order is fixed forever: x first, then y. Always “across, then up”. Figure 1.3 shows how to read these two distances off a point.
Because the order is fixed, (3, 2) and (2, 3) are two different points. The first means “3 across, 2 up”. The second means “2 across, 3 up”. They land in different spots. Mixing up the order is the single most common slip in this whole chapter, so always say it to yourself: across first, then up.
The general rule: for a point P (x, y), the value x is the perpendicular distance from the y-axis, and y is the perpendicular distance from the x-axis.
A small writing tip: people often drop the ”=” sign and write P (x, y) instead of P = (x, y). Both mean the same thing. The short form is handy when you are marking lots of points on a graph.
Are the points (5, 1) and (1, 5) the same point? Why or why not?
How to Plot a Point
“Plotting” a point means: someone gives you a pair like (4, 3), and you mark the matching spot on the grid. It is a two-step walk, and you always start from the origin O.
Let us walk through plotting (4, 3) one step at a time.
Plot the point P (4, 3) on the coordinate plane.
- Start at the origin O, the point (0, 0). This is your home base for every plot.
- Read the first number, the x-coordinate. It is 4 (positive). So walk 4 units to the right along the x-axis. You are now sitting on the x-axis at the mark “4”.
- Read the second number, the y-coordinate. It is 3 (positive). From where you are, climb 3 units straight up.
- Put a dot where you stop and label it P. That dot is the point P (4, 3). (If the x-value had been negative you would walk left; if the y-value had been negative you would go down.)
Figure 1.4 shows this two-step walk as two arrows.
To plot the point (−2, 5), which way do you walk first and how far?
The Four Quadrants
The two axes do more than cross — they slice the whole flat plane into four parts. This plane is called the Cartesian plane, the coordinate plane, or the xy-plane. The four parts are called quadrants (“quad” means four).
They are numbered anticlockwise, starting from the top-right. This numbering is not random — it follows the signs of the coordinates. Figure 1.5 shows all four.
The reason behind each quadrant’s signs is just the rule we already have — right is +, left is −, up is +, down is −:
- Quadrant I (top-right): you are to the right and above O. So (+, +) — both positive.
- Quadrant II (top-left): to the left but still above. So (−, +).
- Quadrant III (bottom-left): to the left and below. So (−, −) — both negative.
- Quadrant IV (bottom-right): to the right but below. So (+, −).
Here is a clean way to see the whole pattern at once.
This table lines up each quadrant with its position and its signs:
| Quadrant | Position | Sign of (x, y) | Example |
|---|---|---|---|
| I | top-right | (+, +) | (4, 2) |
| II | top-left | (-, +) | (-5, 3) |
| III | bottom-left | (-, -) | (-3, -6) |
| IV | bottom-right | (+, -) | (3, -5) |
One more important detail: a point sitting on an axis is not in any quadrant. The axes are the borders between the regions, so points like (4, 0) or (0, −7) belong to the axes themselves, not to a quadrant.
Let us read a quadrant straight off a pair of signs.
Without plotting, say which quadrant the point (−8, −2) lies in.
- Look at the x-coordinate: −8. It is negative, so the point is to the left of the y-axis.
- Look at the y-coordinate: −2. It is negative, so the point is below the x-axis.
- Left and below together means bottom-left. The signs are (−, −).
- The (−, −) region is Quadrant III, so (−8, −2) lies in Quadrant III.
A point has a positive x-coordinate and a negative y-coordinate. Which quadrant is it in?
Distance Between Two Points
Now for the most useful skill in the chapter: measuring the straight-line distance between two points. We build it up in three easy stages.
Stage 1 — points in the same row or same column
If two points lie on the same horizontal line, they share the same y-coordinate. The distance between them is just the gap in their x-values — you subtract.
If two points lie on the same vertical line, they share the same x-coordinate. The distance is the gap in their y-values.
Figure 1.6 shows both cases.
To be safe with negatives, we take the absolute value of the difference (the size of the gap, ignoring any minus sign). In symbols:
Distance between (x₁, y) and (x₂, y) (same row) = |x₂ − x₁|
Distance between (x, y₁) and (x, y₂) (same column) = |y₂ − y₁|
The bars | | mean “make it positive” — a distance can never be negative.
Stage 2 — a slanted segment using Pythagoras
What if the two points are not in the same row or column? Then the line joining them is slanted, and we cannot just subtract. But there is a beautiful trick: build a right-angled triangle and use the Baudhāyana–Pythagoras theorem you met in Class 8.
Because we lean on that theorem here, let us refresh it on the spot.
Here is the idea. Take two points A and D that are not lined up. Draw a horizontal line from one and a vertical line from the other. They meet at a corner, making a right angle. Now the slanted segment AD is exactly the hypotenuse of that right triangle — and its two shorter sides are just the across-gap and the up-gap, which we already know how to find by subtracting. Figure 1.7 does this for A (3, 4) and D (7, 1).
Let us write that calculation out cleanly.
Find the distance AD between A (3, 4) and D (7, 1).
- Find the across-gap (the horizontal side). Subtract the x-coordinates: 7 − 3 = 4 units.
- Find the up-gap (the vertical side). Subtract the y-coordinates: 4 − 1 = 3 units.
- These two are the short sides of a right triangle, and AD is the hypotenuse. Apply Pythagoras: AD² = 4² + 3² = 16 + 9 = 25.
- Take the square root: AD = √25 = 5 units.
A key thing to notice: it does not matter whether the gaps come out positive or negative, because we square them — and squaring kills any minus sign. We are just measuring how far we shifted across and up. This means the formula works even when the points have negative coordinates. (If you reflect the whole triangle across to the negative side, every side keeps the same length — a reflection slides a shape into a mirror image but never stretches or shrinks it.)
Stage 3 — the general distance formula
We can now write one formula that works for any two points. Take A (x₁, y₁) and D (x₂, y₂). The across-gap is (x₂ − x₁) and the up-gap is (y₂ − y₁). Feed them into Pythagoras. Figure 1.8 shows the general triangle.
So here is the formula to remember for the rest of your maths life:
The distance between (x₁, y₁) and (x₂, y₂) is
√( (x₂ − x₁)² + (y₂ − y₁)² )
Let us use it once on points that include negatives, so you trust it fully.
Find the distance between P (−2, −1) and Q (2, 2).
- Label the points: (x₁, y₁) = (−2, −1) and (x₂, y₂) = (2, 2).
- Across-gap: x₂ − x₁ = 2 − (−2) = 2 + 2 = 4. (Subtracting a negative adds.)
- Up-gap: y₂ − y₁ = 2 − (−1) = 2 + 1 = 3.
- Put them in the formula: distance = √(4² + 3²) = √(16 + 9) = √25 = 5 units.
Common Mistakes
These are the slips students make most often. Read each one — knowing the trap is the best way to avoid it.
The point (3, 2) is the same as the point (2, 3).
Both pairs use the same two numbers, 3 and 2, so they look interchangeable — like 3 + 2 and 2 + 3 giving the same total in addition.
Coordinates are an ordered pair. The first number is always the across-distance (x) and the second is always the up-distance (y). (3, 2) means 3 across and 2 up; (2, 3) means 2 across and 3 up. These are two different points.
In the pair (x, y), the first number tells you how far up the point is.
When we plot a point we often look up the page first because the answer is 'up there', so the eye treats the vertical position as the main one and reads it first.
The first number is always x, the across-distance from the y-axis. The second number is y, the up-distance from the x-axis. Across comes first, then up.
A point like (5, 0) or (0, -3) sits inside a quadrant.
The point clearly has a position on the plane, so it feels like it must belong to one of the four regions, just like every other point.
Points with a 0 in their pair sit exactly ON an axis, and the axes are the borders between quadrants. So such points belong to an axis, not to any quadrant.
To find the distance between two slanted points, just add the across-gap and the up-gap.
That is the path length if you walk along the grid lines (across, then up), so it feels like the natural 'distance', and adding is simpler than squaring.
The straight-line distance is the hypotenuse of a right triangle, not the two sides added. You must use √((x2-x1)² + (y2-y1)²). Adding gives the longer L-shaped path, not the direct distance.
A distance can come out negative if the second point has smaller coordinates than the first.
When you subtract a larger number from a smaller one, like 3 - 7, you do get a negative, so it seems the distance could be negative too.
We square each difference in the distance formula, and squaring always gives a non-negative result. Distance measures how far apart two points are, which can never be less than zero.
Quick Check
Test yourself before moving to practice. Try each one before reading the explanation.
In which quadrant does the point (−4, 7) lie?
A point lies on the x-axis, 6 units to the left of the origin. What are its coordinates?
What is the distance between the points (1, 2) and (4, 6)?
Practice Problems
Work through these in order — they go from easy to challenging. Try each fully before revealing the solution.
Easy
Write down the coordinates of the origin. Also write the general form of a point that lies on the x-axis.
The origin is the point where the two axes cross, so its coordinates are (0, 0).
A point on the x-axis is not above or below the axis, so its distance from the x-axis is 0 — that is its y-coordinate. So it has the form (x, 0), where x can be any number.
Name the quadrant or axis for each point: A (3, 5), B (0, −4), C (−2, −6), D (7, −1).
- A (3, 5): both positive (+, +) → Quadrant I.
- B (0, −4): the x-coordinate is 0, so it sits on the y-axis (not in any quadrant).
- C (−2, −6): both negative (−, −) → Quadrant III.
- D (7, −1): positive x, negative y (+, −) → Quadrant IV.
Find the distance between the two points P (2, 3) and Q (2, 9).
Both points have the same x-coordinate (2), so they lie on the same vertical line. The distance is just the gap in the y-values.
Distance = |9 − 3| = 6 units.
(No need for the full formula when the points share a coordinate — just subtract the other one.)
Medium
Think and Reflect: What is the x-coordinate of any point on the y-axis? Is there a similar rule for points on the x-axis?
A point on the y-axis is neither left nor right of the y-axis — its distance from the y-axis is 0. So the x-coordinate of every point on the y-axis is 0. Such points look like (0, y).
By the same reasoning, a point on the x-axis has y-coordinate 0 (its distance from the x-axis is 0). Those points look like (x, 0). So a 0 in either slot tells you which axis the point sits on.
Reiaan's rectangular study table has three of its feet at the points (8, 9), (11, 9) and (11, 7). Where will the fourth foot be? What are the table's width and length?
A rectangle has its corners lined up in rows and columns. Plot the three known feet to see the pattern.
- (8, 9) and (11, 9) share the y-value 9 — they are the top edge.
- (11, 9) and (11, 7) share the x-value 11 — they are the right edge.
The fourth corner must line up below (8, 9) and to the left of (11, 7). So it shares x = 8 (like the first point) and y = 7 (like the third point). The fourth foot is at (8, 7).
- Width (the horizontal side) = |11 − 8| = 3 units.
- Length (the vertical side) = |9 − 7| = 2 units.
We cannot tell the height of the table from this map, because the map only shows the floor (a 2-D top view). Height is the third dimension, which a flat map cannot capture.
Find the lengths of all three sides of the triangle with vertices A (3, 4), D (7, 1) and M (9, 6). Use the distance formula.
Apply distance = √((x₂−x₁)² + (y₂−y₁)²) to each pair.
AD, from (3, 4) to (7, 1): across-gap = 7 − 3 = 4, up-gap = 1 − 4 = −3. AD = √(4² + (−3)²) = √(16 + 9) = √25 = 5 units.
DM, from (7, 1) to (9, 6): across-gap = 9 − 7 = 2, up-gap = 6 − 1 = 5. DM = √(2² + 5²) = √(4 + 25) = √29 units (about 5.39).
MA, from (9, 6) to (3, 4): across-gap = 3 − 9 = −6, up-gap = 4 − 6 = −2. MA = √((−6)² + (−2)²) = √(36 + 4) = √40 units (about 6.32).
Notice the negative gaps did no harm — squaring removed every minus sign.
Challenge
Are the points M (−3, −4), A (0, 0) and G (6, 8) on the same straight line? Suggest a way to check WITHOUT plotting them.
Idea: if three points are on one straight line, then the distance from the first to the last equals the two shorter distances added together (no detour). So compute all three distances and check.
MA, (−3, −4) to (0, 0): √((0−(−3))² + (0−(−4))²) = √(3² + 4²) = √25 = 5.
AG, (0, 0) to (6, 8): √(6² + 8²) = √(36 + 64) = √100 = 10.
MG, (−3, −4) to (6, 8): √((6−(−3))² + (8−(−4))²) = √(9² + 12²) = √(81 + 144) = √225 = 15.
Now check: MA + AG = 5 + 10 = 15 = MG. The two short hops add up exactly to the long one. There is no shortcut between M and G, which can only happen if A lies right on the segment MG. So yes, the three points are collinear (on the same straight line).
Plot A (2, 1), B (−1, 2), C (−2, −1) and D (1, −2) and join them in order. Is ABCD a square? Find its area.
For a square, all four sides must be equal AND the diagonals must be equal. Use the distance formula.
Sides: AB, (2,1)→(−1,2): √((−3)² + 1²) = √(9+1) = √10. BC, (−1,2)→(−2,−1): √((−1)² + (−3)²) = √(1+9) = √10. CD, (−2,−1)→(1,−2): √(3² + (−1)²) = √(9+1) = √10. DA, (1,−2)→(2,1): √(1² + 3²) = √(1+9) = √10.
All four sides equal √10 — so it is at least a rhombus.
Diagonals: AC, (2,1)→(−2,−1): √((−4)² + (−2)²) = √(16+4) = √20. BD, (−1,2)→(1,−2): √(2² + (−4)²) = √(4+16) = √20.
Both diagonals equal √20 — equal diagonals on a rhombus make it a square. So yes, ABCD is a square.
Area of a square = side² = (√10)² = 10 square units.
M (0, 0) is the midpoint of segment ST. The table gives some S and T pairs — for S (−3, 0) and T (3, 0), check whether (0, 0) is the midpoint. Then find a connection between the coordinates of a midpoint M and the endpoints S and T.
The midpoint is the point exactly halfway between two ends. Halfway means its x-value is the average of the two x-values, and its y-value is the average of the two y-values.
Check S (−3, 0) and T (3, 0): average of the x’s = (−3 + 3) ÷ 2 = 0; average of the y’s = (0 + 0) ÷ 2 = 0. That gives (0, 0), which is exactly M. So yes, M is the midpoint here.
The general connection (the midpoint rule): if M is the midpoint of S (x₁, y₁) and T (x₂, y₂), then
M = ( (x₁ + x₂) / 2 , (y₁ + y₂) / 2 )
— just average the x-coordinates and average the y-coordinates. You will use this rule a lot in later coordinate geometry.
Summary
After this chapter, you can now explain and do the following:
- Why one number line is not enough for a flat surface, and how two number lines crossed at right angles (the x-axis and the y-axis) fix any point with a pair of numbers.
- That the crossing point is the origin O = (0, 0), the plane is the Cartesian plane, and a point is named by the ordered pair (x, y) — x is the distance from the y-axis (across), y is the distance from the x-axis (up).
- Why order matters, so (x, y) and (y, x) are different points unless x = y; and why points on the x-axis look like (x, 0) and points on the y-axis look like (0, y).
- How to plot a point by walking from O — across first by the x-value, then up by the y-value (left or down for negatives).
- The four quadrants and their sign patterns: I (+, +), II (−, +), III (−, −), IV (+, −) — numbered anticlockwise — and that points on an axis are in no quadrant.
- How to find the distance between two points: just subtract when they share a coordinate, and otherwise use the Baudhāyana–Pythagoras result √((x₂ − x₁)² + (y₂ − y₁)²).
What’s Next
You have learned to pin down points and shapes using pairs of numbers. The next step is to describe whole lines of points with a single equation — and that is where algebra and geometry shake hands. In Chapter 2 — Introduction to Linear Polynomials, you will meet expressions like 2x + 3, see how they behave, and begin connecting them to the straight lines you can now plot on the coordinate plane.
Frequently Asked Questions
What is the Cartesian coordinate system in simple words?
It is a way of using two numbers to fix the exact spot of any point on a flat surface. You draw two number lines that cross at right angles: a horizontal x-axis and a vertical y-axis. Every point is then named by a pair like (3, 2), telling you how far across and how far up it is from the crossing point. It works just like the grid lines on a map or on graph paper.
How do you find the coordinates of a point on a graph?
Look at the point and drop a straight line down to the x-axis: the number it lands on is the x-coordinate. Then go straight across to the y-axis: that number is the y-coordinate. Write them as a pair with x first and y second, like (3, 2). The x-coordinate is the distance from the y-axis and the y-coordinate is the distance from the x-axis.
Why is the order of coordinates important, so (3, 2) is not the same as (2, 3)?
Because the first number always means the across-distance (x) and the second always means the up-distance (y). So (3, 2) means 3 across and 2 up, while (2, 3) means 2 across and 3 up. These land on two different spots. The only time (x, y) and (y, x) are the same point is when x and y are equal.
In which quadrant is a point with a negative x and positive y?
It lies in Quadrant II, the top-left region. The quadrants are numbered anticlockwise from the top-right: Quadrant I is (+, +), Quadrant II is (-, +), Quadrant III is (-, -), and Quadrant IV is (+, -). So a point like (-5, 3) sits in Quadrant II.
How do you find the distance between two points using the Pythagoras theorem?
Make a right-angled triangle with the two points as the slanted side. The horizontal side is the difference of the x-coordinates, (x2 - x1), and the vertical side is the difference of the y-coordinates, (y2 - y1). By the Baudhayana-Pythagoras theorem the distance is the square root of ((x2-x1)² + (y2-y1)²). For example, the distance between (3, 4) and (7, 1) is √(4² + 3²) = √25 = 5 units.
What are the coordinates of the origin and of points on the axes?
The origin, where the two axes cross, is always (0, 0). Any point on the x-axis has the form (x, 0) because its distance from the x-axis is zero. Any point on the y-axis has the form (0, y) because its distance from the y-axis is zero. So a zero in the pair tells you the point sits exactly on an axis.