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:

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.

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.

Calculate:

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

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Example 1: Finding the Distance Between Two Points Let's find the distance between the points $A(2, 3)$ and $B(5, 7)$ .

Step 1: Label the points:

For point $A$ , $x_1 = 2$ and $y_1 = 3$ .
For point $B$ , $x_2 = 5$ and $y_2 = 7$ .

Step 2: Substitute into the formula:

Start with the distance formula: $\text{Distance } |AB| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Substitute the coordinates from your points, making sure to put the numbers in brackets: $\text{Distance } |AB| = \sqrt{((5) - (2))^2 + ((7) - (3))^2}$

Step 3: Calculate:

Simplify inside the brackets first: $\text{Distance } |AB| = \sqrt{(3)^2 + (4)^2}$
Then square the results: $\text{Distance } |AB| = \sqrt{9 + 16}$
Add the squares together: $\text{Distance } |AB| = \sqrt{25}$
Finally, take the square root of $25$ : $\text{Distance } |AB| = 5$

So, the distance between the points $A$ and $B$ is 5 units.

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:

Label the Points:

Just like with the distance formula, start by labeling 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 $y_2$ .

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 $y-coordinates$ .

Calculate:

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

lightbulbExample

Example 2: Finding the Midpoint Let's find the midpoint of the line segment connecting the points $A(2, 3)$ and $B(8, 7)$ .

Step 1: Label the points:

For point $A$ , $x_1 = 2$ and $y_1 = 3$ .
For point $B,$ $x_2 = 8$ and $y_2 = 7$ .

Step 2: Substitute into the formula:

Start with the midpoint formula: $\text{Midpoint of AB} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
Replace $x_1$ , $y_1$ , $x_2$ , and $y_2$ with the coordinates from your points: $\text{Midpoint of AB} = \left(\frac{(2) + (8)}{2}, \frac{(3) + (7)}{2}\right)$

Step 3: Calculate:

Add the $x-coordinates$ : $2 + 8 = 10$ .
Add the $y-coordinates$ : $3 + 7 = 10$ .
Divide each result by $2$ to find the midpoint: $\text{Midpoint} = \left(\frac{10}{2}, \frac{10}{2}\right) = (5, 5)$

So, the midpoint of the line segment $AB$ is (5, 5).

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Key Tips for Success

Label Carefully: Always start by labeling your points clearly as $x_1$ , $y_1$ , $x_2$ , and $y_2$ . This ensures you substitute the correct numbers into the formulas.
Substitute in Brackets: When plugging in values, especially negatives, use brackets to avoid mistakes.
Show Your Work: Write down every step when you substitute the values into the formulas. This helps you see where each number comes from and makes it easier to catch mistakes.
Use a Calculator: When working with larger numbers or square roots, use a calculator to make sure your answers are accurate.

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.

Distance and Midpoint Formula Simplified Revision Notes for Junior Cycle Mathematics

Distance and Midpoint Formula

Distance Formula

Midpoint Formula

Key Tips for Success

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