Distance Formula

The distance formula helps us find the straight-line distance between two points in the Cartesian plane. Instead of counting squares on the graph, we use a formula based on the coordinates.

If the coordinates of two points are known, we can calculate how far apart they are using a simple method.

To understand the distance formula, imagine two points:

\(A(x_1, y_1)\) and \(B(x_2, y_2)\)

If we draw a horizontal and vertical line through these points, they form a right-angled triangle.

• The horizontal length = \(|x_2 - x_1|\)
• The vertical length = \(|y_2 - y_1|\)

Using the Pythagoras Theorem:

Hypotenuse² = base² + height²

Using Pythagoras Theorem, the distance between two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is:

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

This gives the direct (shortest) distance between the two points.

Sometimes the points lie on the same line, which makes calculations easier.

Here are some examples to understand the formula better.

• Forgetting to square the differences.
• Mixing up x and y values.
• Ignoring negative signs while subtracting.
• Taking square root incorrectly.

Find the distance between the following pairs of points:

1. (1, 2) and (4, 6)
2. (-2, -3) and (3, 4)
3. (0, 5) and (0, -3)
4. (7, 1) and (-1, 1)

• The distance formula is based on the Pythagoras Theorem.
• Formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
• Useful for finding shortest distance between two points.
• Special cases simplify the formula when x or y coordinates match.