Line segments (AQA GCSE Maths): Revision Notes
Line segments
What is a line segment?
A line segment represents the straight distance between any two specific points on a coordinate plane. Unlike a full line that extends infinitely in both directions, a line segment has definite start and end points.
To find the exact length of a line segment when you know the coordinates of both endpoints, you can use Pythagoras' theorem. This works because the line segment becomes the hypotenuse of a right-angled triangle.
The key difference between a line and a line segment is that a line extends infinitely in both directions, while a line segment has two specific endpoints and a measurable length.
Method for calculating line segment length
When you have two points with coordinates, follow these steps:
- Sketch the axes and mark both points clearly on your coordinate grid
- Draw a right-angled triangle with the line segment as the hypotenuse
- Calculate the horizontal distance by subtracting the x-coordinates
- Calculate the vertical distance by subtracting the y-coordinates
- Apply Pythagoras' theorem to find the length of the hypotenuse
- Round your answer to the required number of decimal places
The calculation process
For any two points with coordinates and :
- Horizontal distance =
- Vertical distance =
- Length =
The absolute value symbols (| |) ensure that distances are always positive, regardless of which point has larger coordinates.
Worked example
Worked Example: Finding Line Segment Length
Let's find the length of line segment AB where:
- Point A has coordinates (2, 5)
- Point B has coordinates (3, -2)
Step 1: Calculate the distances
- Horizontal distance =
- Vertical distance =
Step 2: Apply Pythagoras' theorem
- (to 2 decimal places)
Important exam tips
Common Mistakes to Avoid:
-
Watch out for negative coordinates - always be careful when subtracting negative numbers. Remember that subtracting a negative number is the same as adding a positive number.
-
Use absolute values - since you're calculating distances, the result should always be positive. If you get a negative result, use the absolute value.
-
Check your decimal places - exam questions will specify how many decimal places to give in your answer. Make sure you round correctly.
-
Show your working - always show each step clearly, especially the Pythagoras' theorem calculation.
Key takeaways
Remember These Key Points:
- A line segment is the straight distance between two specific points
- Use Pythagoras' theorem to calculate the length: create a right-angled triangle with the line segment as the hypotenuse
- Calculate horizontal and vertical distances separately, then apply the theorem
- Be extra careful with negative coordinates - subtract them correctly
- Always round your final answer to the specified number of decimal places