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Friday 23rd of June
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# The Distance Formula

The distance between any two points in the xy-plane can be found using the distance formula.

Formula

The Distance Formula

Let (x1, y1) and (x2, y2) represent any two points in the xy-plane. The distance, d, between the points is given by

If the points lie on a horizontal line, then this simplifies to d = | x2 - x1 |.

If the points lie on a vertical line, then this simplifies to d = |y2 - y1|.

Note:

In the distance formula, it doesnâ€™t matter which point is considered (x1, y1) or (x2, y2). The resulting distance is the same.

Example 1

Find the distance between (-3, 4) and (5, 4).

Solution

Since the points (-3, 4) and (5, 4) have the same y-coordinate, 4, they lie on a horizontal line.

 So, we use the formula: Let x1 =-3 and x2 = 5. Simplify. Find the absolute value. The distance between (-3, 4) and (5, 4) is 8 units. d = | x2 - x1 |d = | 5 - (-3) | d = | 8 | d = 8

In the formula, d = | x2 - x1 |, it doesnâ€™t matter which x-coordinate is assigned to x1 and which is assigned to x2. The resulting distance is the same. That is:

|5 - (-3)| = |-3 - 5|

|8| = |-8|

8 = 8

Example 2

Find the distance between (-3, 8) and (5, -1).

Solution

 The points do not lie on a horizontal or vertical line. Therefore, use the distance formula. Let (x1, y1) = (-3, 8) and (x2, y2) = (5, -1). Substitute. Simplify.

The distance between the points (-3, 8) and (5, -1) is units.

Note:

If we switch the points and let (x1, y1) = (5, -1) and (x2, y2) = (-3, 8), we get the same answer.

We can use a calculator to approximate ≈ 12.04.