Calculating the Distance Between Two Points Using the Distance Formula

In coordinate geometry, the distance formula helps us to find the distance between two points in a plane. The formula is derived from the Pythagorean theorem and is an essential tool for various applications, ranging from basic geometry to advanced calculus. The distance formula is given by:

Distance Formula

The distance, d, between two points A(x1, y1) and B(x2, y2) in a plane is calculated as:

d √[(x2 - x1)2 (y2 - y1)2]

Example Calculation: Distance Between Points A(-8, 4) and B(4, -1)

In this example, we will calculate the distance between the points A(-8, 4) and B(4, -1).

Given:

A(x1, y1) (-8, 4) B(x2, y2) (4, -1)

Substitute the given points into the distance formula:

d √[(4 - (-8))2 (-1 - 4)2]

Perform the operations inside the square root:

d √[(4 8)2 (-1 - 4)2]

d √[122 (-5)2]

d √[144 25]

d √169

d 13

Therefore, the distance between points A(-8, 4) and B(4, -1) is 13 units.

Explanation of the Steps

Step 1: Substitute the coordinates of points A and B into the formula. Here, x1 -8, y1 4, x2 4, and y2 -1.

Step 2: Calculate the differences (x2 - x1) and (y2 - y1).

Since x2 - x1 4 - (-8) 4 8 12, and y2 - y1 -1 - 4 -5, the formula becomes:

d √[122 (-5)2]

Step 3: Square the differences and add them together.

122 144, and (-5)2 25.

d √(144 25) √169 13

Thus, the distance is 13 units.

Key Points to Remember

The distance formula is a useful tool in coordinate geometry. The formula can be applied to any pair of points in a plane. Ensure to correctly substitute the coordinates into the formula. Pay attention to the signs of the coordinates when calculating the differences.

By using the distance formula, you can easily calculate the distance between any two points in a coordinate plane, making it a valuable skill for various mathematical and real-world applications.