Line Segment Properties: Midpoint, Length, and Gradient

1. Gradient: Measuring the Steepness of a Line

The gradient of a line is simply a number that tells us how steep it is.

• A positive gradient means the line slopes upwards from left to right.
• A negative gradient means it slopes downwards.
• A gradient of zero means the line is horizontal.

To see a quick visual of this, watch the first part of the video below.

The Formula for Gradient

To calculate the gradient ($m$) between two points, $(x_1,y_1)$ and $(x_2,y_2)$, we use the formula:

$m=\frac{\text{change in }y}{\text{change in }x}=\frac{y_2-y_1}{x_2-x_1}$

Crucial Tip for Accuracy: To avoid the small arithmetic errors you've mentioned, always label your points before you start. For points A($x_1,y_1$) and B($x_2,y_2$), write the labels directly above the numbers in the question. It makes substituting into the formula much safer. It doesn't matter which point is 1 and which is 2, as long as you are consistent in the numerator and denominator.

Let's see this formula in action.

2. Midpoint: Finding the Exact Centre

The midpoint of a line segment is the point that lies exactly halfway between the two endpoints. The easiest way to think about this is that you are finding the average of the x-coordinates and the average of the y-coordinates.

The Formula for Midpoint

The formula for the midpoint ($M$) of a line segment joining $(x_1,y_1)$ and $(x_2,y_2)$ is:

$M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Notice that the final answer is a coordinate pair $(x,y)$.

Now, let's watch a quick worked example.

3. Length: Calculating the Distance Between Two Points

Finding the length of a line segment is a direct application of a familiar concept from your physics and maths background: Pythagoras's Theorem.

If you imagine a right-angled triangle where the line segment is the hypotenuse, the other two sides are simply the "change in x" and the "change in y".

The Formula for Length

The formula for the length ($d$, for distance) of a line segment between $(x_1,y_1)$ and $(x_2,y_2)$ is derived directly from Pythagoras's theorem $a^2+b^2=c^2$:

$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$

This formula might look complex, but it's just $\sqrt{(\Delta x{)}^2+(\Delta y{)}^2}$.

Let's watch a video that demonstrates this visual link and then applies the formula.

4. Application and Practice

Now it's time to put it all together and practice, which is key to your "Error-Loop Method". First, let's watch one comprehensive past paper question being solved, covering all three skills.

Now it's your turn. Use the following resource to access past paper questions. Your task is to apply the methods we've just covered.

Conclusion

Great job today. You have established a solid foundation in coordinate geometry, a topic that carries significant marks and builds throughout the syllabus. By mastering these three core calculations, you are in a much stronger position.

Key Takeaways:

• Gradient: Steepness, calculated with $m=\frac{y_2-y_1}{x_2-x_1}$.
• Midpoint: The average of the points, calculated with $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.
• Length: Distance, calculated using Pythagoras's theorem: $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$.
• Technique: To improve accuracy, always label your coordinates $(x_1,y_1)$ and $(x_2,y_2)$ before substituting them into any formula.

Preview of the Next Lesson:
In our next lesson, we will build directly on today's work. You will learn how to find the equation of a straight line ($y=mx+c$). You already know how to find the gradient, 'm'—the next step is to use that to find the full equation.