Distance/Midpoint Formula (Junior Cert Mathematics): Revision Notes

Distance and Midpoint Formula

In this section, we will learn how to find the distance between two points on the Cartesian Plane and how to find the midpoint of a line segment connecting two points. We'll take it step by step to make sure everything is clear and easy to understand.

Distance Formula

The distance formula is used to find out how far apart two points are on the Cartesian Plane. The formula might look a bit complicated, but don't worry—once you understand how to use it, it's quite straightforward!

The formula is:

$\text{Distance } |AB| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

This formula tells you how to calculate the distance between two points, $A$ and $B$, which have coordinates $A(x_1, y_1)$ and $B(x_2, y_2)$.

Let's break this down into easy steps:

1. Label the Points:

• The first point, $A$, has coordinates $x_1$ and $y_1$.
• The second point, $B$, has coordinates $x_2$ and $y_2$. It's very important to label your points correctly. Always make sure you're consistent: $x_1$ and $y_1$ come from the first point, and $x_2$ and $y_2$ come from the second point.

2. Substitute into the Formula:

• When substituting the numbers into the formula, be sure to put the numbers in brackets, especially if they are negative. This helps avoid mistakes.
• Plug the values of $x_1$, $y_1$, $x_2$, and $y_2$ into the formula.

3. Calculate:

• Do the maths in one go, following the order of operations (brackets, exponents, then addition). This will give you the final distance.

Midpoint Formula

The midpoint formula helps you find the point that's exactly halfway between two points on a line segment. This point is called the midpoint. The formula for finding the midpoint is:

$\text{Midpoint of AB} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Here's how to use it step by step:

4. Label the Points:

• Just like with the distance formula, start by labelling the coordinates of the two points. Let's call the first point $A$ with coordinates $x_1$ and $y_1$, and the second point $B$ with coordinates $x_2$ and $y_2$.

5. Substitute into the Formula:

• Plug the coordinates into the midpoint formula. This means you'll add the $x-coordinates$ together and divide by 2 to find the $x-coordinate$ of the midpoint. Then, do the same for the $y-coordinates$.

6. Calculate:

• Find the average of the $x-coordinates$ and the $y-coordinates$. This gives you the coordinates of the midpoint.

These formulas are essential tools in co-ordinate geometry. With practice, you'll get more comfortable using them, and they will become second nature when solving problems.