Introduction

In the previous lesson, you learned how to plot and name points using ordered pairs. Once points are plotted, we can describe their locations more precisely by identifying the quadrant or axis where they lie.

The x-axis and y-axis divide the coordinate plane into four sections, called quadrants. Each quadrant has its own combination of positive and negative coordinates.

Understanding quadrants helps us quickly identify the location of any point on the coordinate plane.

 

5

The Four Quadrants

The x-axis and y-axis divide the coordinate plane into four regions.

The quadrants are numbered using Roman numerals, starting in the upper-right corner and moving counterclockwise.

The numbering always follows this order:

• Quadrant I
• Quadrant II
• Quadrant III
• Quadrant IV

Remembering the Signs

Each quadrant has a different combination of positive and negative numbers.

Many students remember the pattern by starting in Quadrant I and moving counterclockwise.

 

 

5

Identifying the Quadrant of a Point

To determine which quadrant a point belongs to, simply examine the signs of its coordinates.

Example 1

Point:

(4, 3)

• x is positive
• y is positive

Therefore, the point is in Quadrant I.

Example 2

Point:

(-5, 2)

• x is negative
• y is positive

This point is in Quadrant II.

Example 3

Point:

(-4, -6)

Both coordinates are negative.

The point is in Quadrant III.

Example 4

Point:

(7, -3)

• x is positive
• y is negative

This point lies in Quadrant IV.

Points on the Axes

Not every point belongs to a quadrant.

Some points lie directly on an axis.

Points on the x-axis

If the y-coordinate is zero, the point lies on the x-axis.

Examples:

• (4,0)
• (-7,0)
• (0,0)

Points on the y-axis

If the x-coordinate is zero, the point lies on the y-axis.

Examples:

• (0,5)
• (0,-3)
• (0,8)

The Origin

The point

(0,0)

is called the origin.

The origin belongs to both axes, but not to any quadrant.

 

 

5

Classifying Points

Every point can be classified by its location.

Real-World Applications

Coordinates and quadrants are used in many fields where precise locations are important.

 

5

Examples include:

• GPS navigation
• Robotics
• Engineering
• Architecture
• Computer graphics
• Video game design
• Geographic Information Systems (GIS)
• Scientific data mapping

Worked Examples

Example 1

Determine the quadrant of:

(-8,4)

• x is negative
• y is positive

Answer: Quadrant II

Example 2

Determine the quadrant of:

(6,-5)

• x is positive
• y is negative

Answer: Quadrant IV

Example 3

Where is the point (0,7)?

The x-coordinate is zero.

Answer: On the y-axis

Example 4

Where is the point (-9,0)?

The y-coordinate is zero.

Answer: On the x-axis

Example 5

Where is the point (0,0)?

Both coordinates are zero.

Answer: The origin

Did You Know?

Pilots and air traffic controllers use coordinate systems to monitor the exact positions of aircraft. Similar coordinate systems are also used by ships, satellites, and space missions to navigate accurately over long distances.

Key Terms

Key Takeaways

• The coordinate plane is divided into four quadrants.
• Each quadrant has a unique combination of positive and negative coordinates.
• Quadrants are numbered I, II, III, and IV in a counterclockwise direction.
• Points with y = 0 lie on the x-axis.
• Points with x = 0 lie on the y-axis.
• The point (0,0) is called the origin and does not belong to any quadrant.

Introduction

Symmetry is everywhere around us. Butterflies, flowers, buildings, snowflakes, and even human faces often show symmetry. In mathematics, symmetry describes shapes or figures that can be divided into matching halves.

On a coordinate plane, symmetry is closely related to reflections. A reflection creates a mirror image of a point or shape across a line, such as the x-axis or y-axis.

Understanding reflections helps us solve geometric problems and recognize patterns in mathematics.

6

What is Symmetry?

A figure has symmetry if it can be divided into two matching halves.

The dividing line is called the line of symmetry.

If one half is folded over the line of symmetry, it matches the other half exactly.

Examples of symmetrical objects include:

• Butterflies
• Leaves
• Snowflakes
• Many company logos
• Some letters of the alphabet

Lines of Symmetry

A line of symmetry can be:

• Vertical
• Horizontal
• Diagonal

Some shapes have several lines of symmetry.

Examples:

Reflections on the Coordinate Plane

A reflection flips a point or shape across a line, producing a mirror image.

The most common lines of reflection are:

• the x-axis
• the y-axis

The reflected point remains the same distance from the line of reflection as the original point.

Reflection Across the x-axis

When a point is reflected across the x-axis:

• the x-coordinate stays the same
• the y-coordinate changes sign

Rule: (x,y)→(x,−y)

Example

Original point:

(4,3)

Reflected across the x-axis:

(4,-3)

The point moves the same distance below the x-axis.

5

Reflection Across the y-axis

When a point is reflected across the y-axis:

• the y-coordinate stays the same
• the x-coordinate changes sign

Rule: (x,y)→(−x,y)

Example

Original point:

(5,-2)

Reflected point:

(-5,-2)

The point moves the same distance to the opposite side of the y-axis.

 

5

Reflection Across Both Axes

If a point is reflected across both the x-axis and y-axis:

• both coordinates change sign.

Rule: (x,y)→(−x,−y)

Example

Original point:

(6,4)

Reflected point:

(-6,-4)

This places the point in the diagonally opposite quadrant.

Recognizing Reflection Patterns

Notice the patterns:

A useful way to remember the rules is:

• Reflect across the x-axis → change y
• Reflect across the y-axis → change x

Reflecting Shapes

Reflections are not limited to single points.

Entire shapes can be reflected.

To reflect a polygon:

1. Reflect each vertex.
2. Plot the new coordinates.
3. Join the reflected vertices in the same order.

 

6

Real-World Applications

Reflections and symmetry are used in many fields.

 

 

 

7

Applications include:

• Architecture
• Art and design
• Engineering
• Computer graphics
• Animation
• Robotics
• Photography
• Logo design

Many famous buildings use symmetry because people naturally find symmetrical designs attractive and balanced.

Worked Examples

Example 1

Reflect (5,2) across the x-axis.

Solution:

• x stays the same.
• y changes sign.

Answer: (5,-2)

Example 2

Reflect (-3,6) across the y-axis.

Solution:

• x changes sign.
• y stays the same.

Answer: (3,6)

Example 3

Reflect (-4,-7) across both axes.

Solution:

Both coordinates change sign.

Answer: (4,7)

Example 4

Which reflection changes only the y-coordinate?

Answer: Reflection across the x-axis

Example 5

Which reflection changes only the x-coordinate?

Answer: Reflection across the y-axis

Did You Know?

Many species of animals, including butterflies, birds, and humans, show bilateral symmetry, meaning their left and right sides are nearly mirror images. Scientists believe symmetry often plays an important role in movement, balance, and even mate selection in nature.

Key Terms

Key Takeaways

• A symmetrical figure has matching halves separated by a line of symmetry.
• Reflections create mirror images across a line.
• Reflecting across the x-axis changes the y-coordinate.
• Reflecting across the y-axis changes the x-coordinate.
• Reflecting across both axes changes the signs of both coordinates.
• Reflections are widely used in geometry, art, architecture, engineering, and computer graphics.

Introduction

Coordinate systems are not just used in mathematics classrooms—they are an essential part of modern life. Every day, people use coordinates without even realizing it. Whether you are following directions on your phone, finding a seat in a stadium, playing a video game, or locating a city on a map, you are using a coordinate system.

Coordinate systems allow us to describe exact locations and track movement with precision. They are used in navigation, engineering, science, transportation, and many forms of technology.

What is a Real-World Coordinate System?

A coordinate system is a method of describing the location of an object using numbers.

In mathematics, we use the Cartesian coordinate plane.

In everyday life, many different coordinate systems are used depending on the situation.

Some examples include:

• Street maps
• GPS navigation
• Flight navigation
• Video game maps
• Weather maps
• Architectural plans
• Sports fields
• Computer graphics

Although they may look different, they all have the same purpose:

To describe locations accurately.

Maps and Grid References

Many maps are divided into square grids.

Each square has a unique reference that helps people locate places quickly.

For example, a city map might use:

• A–F across the top
• 1–8 down the side

A location at C4 can be found quickly by following the grid.

 

6

Maps with grid references are commonly used by:

• Emergency services
• Delivery drivers
• Tourists
• Military personnel
• Hikers

GPS Coordinates

One of the most important coordinate systems today is the Global Positioning System (GPS).

GPS uses a network of satellites orbiting Earth.

Your phone communicates with several satellites to calculate your exact position.

Instead of x- and y-coordinates, GPS uses:

• Latitude
• Longitude

Latitude measures how far north or south a location is from the Equator.

Longitude measures how far east or west a location is from the Prime Meridian.

Together, these coordinates identify almost any location on Earth.

5

Navigation Systems

Modern navigation systems constantly calculate your position.

Examples include:

• Car navigation systems
• Smartphone map apps
• Aircraft navigation
• Ship navigation
• Drone navigation

As you move, your coordinates change continuously.

The navigation system compares your current location with your destination to calculate:

• Distance
• Direction
• Estimated travel time
• Best route

Coordinates in Video Games

Many video games use coordinate systems to determine the position of every object.

Characters, buildings, enemies, and items all have coordinates.

Some games even display your location.

For example:

Player Position:

(145, 82)

This tells the game exactly where the player is located.

 

6

Game designers use coordinates to:

• Move characters
• Detect collisions
• Create maps
• Position objects
• Control animations

Coordinates in Computer Graphics

Everything displayed on a computer screen has a position.

Each pixel has coordinates.

Graphic designers use coordinate systems to place:

• Images
• Buttons
• Text
• Icons
• Animations

Computer programmers also use coordinates when creating websites, apps, and games.

Coordinates in Science and Engineering

Scientists and engineers rely on coordinate systems every day.

Applications include:

• Designing buildings
• Planning roads
• Creating bridges
• Mapping the ocean floor
• Surveying land
• Designing robots

7

Without coordinate systems, it would be impossible to build structures accurately.

Coordinates in Sports

Many sports use coordinate systems to analyse player movement.

Examples include:

• Football player tracking
• Tennis ball trajectories
• Basketball shot locations
• Cricket ball tracking
• Baseball pitch analysis

Professional teams collect coordinate data to improve tactics and performance.

Representing Movement

Coordinates can also describe movement.

Suppose a robot starts at:

(2,3)

It moves:

• 4 units right
• 2 units up

Its new position is:

(6,5)

By recording coordinates at different times, we can track the path of moving objects such as:

• Cars
• Drones
• Animals
• Spacecraft
• Robots

Practical Example

Imagine you are delivering a package in a large warehouse.

The warehouse uses a coordinate grid.

The package is stored at:

(12,8)

Instead of searching every shelf, workers simply follow the coordinates directly to the correct location.

Large warehouses operated by online retailers often use this type of system to locate thousands of products quickly and accurately.

Real-World Applications

 

 

4

Coordinate systems are used in:

• GPS navigation
• Google Maps
• Aviation
• Shipping
• Robotics
• Engineering
• Surveying
• Architecture
• Emergency services
• Weather forecasting
• Computer graphics
• Video game development
• Scientific research
• Warehouse management

Worked Examples

Example 1

A city map uses a grid.

The library is located at D5.

How would you find it?

Answer:

Move across to column D, then move down to row 5.

Example 2

A robot moves from (3,2) to (7,5).

How far did it move?

Answer:

• 4 units to the right
• 3 units upward

Example 3

A drone starts at (10,6).

It flies 5 units west and 2 units south.

New coordinates:

(5,4)

Example 4

A delivery driver follows GPS directions.

Why are coordinates useful?

Answer:

They provide an exact location, making navigation faster and more accurate.

Did You Know?

More than 30 GPS satellites orbit Earth, allowing navigation devices almost anywhere on the planet to determine their positions. Modern smartphones combine GPS with Wi-Fi and mobile network signals to improve location accuracy, especially in cities and indoors.

Key Terms

Key Takeaways

• Coordinate systems are used to describe precise locations.
• Maps, GPS devices, and navigation systems all rely on coordinates.
• Coordinates help track movement and determine positions.
• Many technologies, including smartphones, aircraft, robots, and video games, depend on coordinate systems.
• Understanding coordinate geometry helps solve practical problems in everyday life, science, engineering, and technology.