Introduction to Coordinates

Q: What are the key components of the rectangular coordinate system (Cartesian plane)?

The Cartesian plane consists of a horizontal x-axis , a vertical y-axis , and their intersection called the origin extbf{(0, 0)} .

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Mathematics · Junior Secondary · 7 min read · Updated: Aug 17, 2026

Introduction to Coordinates: Your Guide to Mapping the Maths World!

Hello there! Welcome to the amazing world of coordinates. Ever played a video game where you have to find a character on a map? Or used a map app to find your way? That's all thanks to coordinates!

This chapter is all about learning how to describe the exact location of a point on a flat surface, called a plane. It's like learning the secret code to a treasure map. By the end of this, you'll be able to plot points in rectangular and polar coordinates, find distances between horizontal and vertical points, calculate areas, and transform shapes using numbers. Let's begin our adventure!

1. The Rectangular Coordinate System: Our Map

To find locations, we need a map. In maths, our map is called the Rectangular Coordinate System or the Cartesian Plane. It looks like a grid with two main lines crossing each other at a right angle.

The Key Parts of Our Map

The x-axis: This is the horizontal line that goes left and right. Think of it as a corridor you walk along.
The y-axis: This is the vertical line that goes up and down. Think of it as the stairs you climb up or down.
The Origin: This is the special point where the x-axis and y-axis cross. It's our starting point, and its address is always (0, 0).

Coordinates: The Address of a Point

Every point on the plane has a unique address called its coordinates, which we write as an ordered pair: \((x, y)\).

The first number is the x-coordinate. It tells you how far to move left or right from the origin.
The second number is the y-coordinate. It tells you how far to move up or down.

Memory Aid: Run Before You Jump!

A simple way to remember the order \((x, y)\) is: You have to 'run' (along the horizontal x-axis) before you can 'jump' (up or down the vertical y-axis). Also, X comes before Y in the alphabet!

Example: Let's find the point \(P(3, 2)\).

1. Start at the Origin \((0, 0)\).
2. 'Run' 3 units to the right along the x-axis (because the x-coordinate is positive 3).
3. From there, 'Jump' 2 units up (because the y-coordinate is positive 2).
4. Mark that spot. That's point P!

The Four Quadrants

The x and y axes divide the plane into four sections called quadrants (numbered anti-clockwise starting from the top-right). The signs of the coordinates tell you which quadrant a point is in:

Quadrant I: Top right. Here, both x and y are positive: \((+, +)\)
Quadrant II: Top left. Here, x is negative and y is positive: \((-, +)\)
Quadrant III: Bottom left. Here, both x and y are negative: \((-, -)\)
Quadrant IV: Bottom right. Here, x is positive and y is negative: \((+, -)\)

Key Takeaway

The rectangular coordinate system gives every point a unique \((x, y)\) address, telling us exactly where it is on the plane.

2. The Polar Coordinate System

Besides the rectangular grid, there is another popular way to describe the location of a point: the Polar Coordinate System.

Instead of moving along a grid of horizontal and vertical lines, polar coordinates describe a point using a distance from a fixed reference point and an angle from a fixed direction.

The Key Parts of Polar Coordinates

The Pole: The fixed starting point (like the origin \(O\)).
The Polar Axis: A horizontal ray pointing directly to the right from the pole (similar to the positive x-axis).
Coordinates \((r, \theta)\):
- \(r\) is the straight-line distance from the pole to the point (where \(r \ge 0\)).
- \(\theta\) is the angle measured anti-clockwise from the polar axis to the line connecting the pole and the point.

Example: Plot the point \(A(4, 60^\circ)\).

1. Start at the polar axis.
2. Rotate anti-clockwise by an angle of \(60^\circ\).
3. Move a distance of 4 units outward along this direction to plot point A.

Key Takeaway

In polar coordinates \((r, \theta)\), \(r\) gives the distance from the pole and \(\theta\) gives the anti-clockwise direction angle.

3. Finding Simple Distances

Let's find the distance between points on a straight horizontal or vertical line.

Distance Between Two Points on a Horizontal Line

If two points lie on the same horizontal line, their y-coordinates are identical. To find the distance between them, calculate the difference between their x-coordinates:

Distance = \(|x_2 - x_1|\) (the positive difference between \(x_1\) and \(x_2\))

Example: Find the distance between \(A(2, 3)\) and \(B(7, 3)\).

1. Notice both y-coordinates are 3 (horizontal line).
2. Subtract the x-coordinates: \(7 - 2 = 5\).
3. The distance between A and B is 5 units.

Distance Between Two Points on a Vertical Line

If two points lie on the same vertical line, their x-coordinates are identical. To find the distance between them, calculate the difference between their y-coordinates:

Distance = \(|y_2 - y_1|\)

Example: Find the distance between \(C(4, 1)\) and \(D(4, 6)\).

1. Notice both x-coordinates are 4 (vertical line).
2. Subtract the y-coordinates: \(6 - 1 = 5\).
3. The distance between C and D is 5 units.

Key Takeaway

For points on a horizontal or vertical line, distance is simply the positive difference between the non-identical coordinates.

4. Finding Areas of Simple Polygons

We can combine our knowledge of plotting points and finding horizontal/vertical distances to calculate the areas of polygons on the Cartesian plane.

Method 1: Direct Formula (Base and Height)

For rectangles, parallelograms, and triangles with horizontal or vertical sides, find the side lengths directly and apply the area formula.

Example: Find the area of rectangle ABCD with vertices \(A(1, 1)\), \(B(7, 1)\), \(C(7, 4)\), and \(D(1, 4)\).

1. Base \(AB\) is horizontal: \(\text{Length} = |7 - 1| = 6\) units.
2. Height \(BC\) is vertical: \(\text{Length} = |4 - 1| = 3\) units.
3. \(\text{Area} = \text{Base} \times \text{Height} = 6 \times 3 = 18\) sq. units.

Method 2: Splitting or Box Method

For irregular rectilinear figures, split the shape into simple rectangles and triangles, or enclose the figure in a large rectangle and subtract the outer unshaded areas.

Key Takeaway

To find the area of a polygon on the coordinate plane, measure horizontal and vertical segments directly and apply standard geometric area formulas.

5. Transformations on the Coordinate Plane

A transformation moves or changes the position of a point or shape on the coordinate plane. Let's look at translation, reflection, and rotation.

1. Translation (Sliding)

Translation slides a point horizontally and/or vertically without changing its orientation:

Move right by \(a\) units: \((x, y) \rightarrow (x + a, y)\)
Move left by \(a\) units: \((x, y) \rightarrow (x - a, y)\)
Move up by \(b\) units: \((x, y) \rightarrow (x, y + b)\)
Move down by \(b\) units: \((x, y) \rightarrow (x, y - b)\)

Example: Translate \(A(3, 4)\) by 2 units left and 5 units down:

New \(x = 3 - 2 = 1\), new \(y = 4 - 5 = -1\). The new point is \(A'(1, -1)\).

2. Reflection (Flipping)

Reflection creates a mirror image across a line of reflection:

Across the x-axis: \((x, y) \rightarrow (x, -y)\)
Across the y-axis: \((x, y) \rightarrow (-x, y)\)
Across a vertical line \(x = c\): The y-coordinate remains unchanged, and the point remains the same distance from the mirror line on the other side.
Across a horizontal line \(y = c\): The x-coordinate remains unchanged, and the point remains the same distance from the mirror line on the other side.

Example: Reflect \(P(1, 3)\) across the vertical line \(x = 4\).

The distance from \(x = 1\) to \(x = 4\) is 3 units to the right. Moving another 3 units to the right gives \(x = 4 + 3 = 7\). The reflected point is \(P'(7, 3)\).

3. Rotation about the Origin (Turning)

Turning a point around the origin \((0, 0)\) by a standard angle (anti-clockwise):

\(90^\circ\) anti-clockwise (or \(270^\circ\) clockwise): \((x, y) \rightarrow (-y, x)\)
\(180^\circ\) anti-clockwise (or \(180^\circ\) clockwise): \((x, y) \rightarrow (-x, -y)\)
\(270^\circ\) anti-clockwise (or \(90^\circ\) clockwise): \((x, y) \rightarrow (y, -x)\)

Example: Rotate \(P(2, 3)\) by \(90^\circ\) anti-clockwise about the origin.

Using the rule \((x, y) \rightarrow (-y, x)\), the new point is \(P'(-3, 2)\).

Key Takeaway

Transformations map points to new coordinates using precise algebraic rules for sliding (translation), flipping (reflection across axes or parallel lines), and turning (rotation about the origin).

References

Supplement to Mathematics Education Key Learning Area Curriculum Guide: Learning Content of Junior Secondary Mathematics (2017) — Curriculum Development Council, Education Bureau, HKSAR
Mathematics Education – Curriculum Documents — Education Bureau (EDB), HKSAR
Web-based Learning and Teaching Support (WLTS): Junior Secondary Mathematics Learning Objectives — Education Bureau, HKSAR

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What are the key components of the rectangular coordinate system (Cartesian plane)?

The Cartesian plane consists of a horizontal x-axis, a vertical y-axis, and their intersection called the origin extbf{(0, 0)}.

How is a point's position represented in rectangular coordinates?

A point is represented as an ordered pair extbf{\((x, y)\)}, where \(x\) indicates horizontal distance from the origin and \(y\) indicates vertical distance.

What are the coordinate sign patterns of the four quadrants in a Cartesian plane?

Quadrant I is \((+, +)\), Quadrant II is \((-, +)\), Quadrant III is \((-, -)\), and Quadrant IV is \((+, -)\).

How does the Polar Coordinate System define the position of a point?

Polar coordinates use \((r, \theta)\), where \(r\) is the distance from a fixed pole and \(\theta\) is the anti-clockwise angle from the polar axis.

How do you find the distance between two points that lie on the same horizontal line?

The distance is the absolute difference between their x-coordinates: \(\text{Distance} = |x_2 - x_1|\).

How do you find the distance between two points that lie on the same vertical line?

The distance is the absolute difference between their y-coordinates: \(\text{Distance} = |y_2 - y_1|\).

How are the areas of simple polygons calculated on the coordinate plane?

Find horizontal and vertical side lengths from coordinate differences and apply standard area formulas, or use the splitting/box method for irregular figures.

What are the coordinate rules for translating a point horizontally and vertically?

Translating right/left by \(a\) and up/down by \(b\) maps \((x, y) \rightarrow (x \pm a, y \pm b)\).

What are the coordinate rules for reflecting a point across the coordinate axes?

Reflection across the x-axis gives \((x, -y)\), and reflection across the y-axis gives \((-x, y)\).

What are the coordinate rules for rotating a point \((x, y)\) anti-clockwise about the origin?

\(90^\circ\): \((x, y) \rightarrow (-y, x)\); \(180^\circ\): \((x, y) \rightarrow (-x, -y)\); \(270^\circ\): \((x, y) \rightarrow (y, -x)\).

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