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The Distance and Midpoint Formulas

Recall from the Pythagorean Theorem that, in a right triangle, the hypotenuse c and sides a and b are related by a2 + b2 = c2. Conversely, if a2 + b2 = c2, the triangle is a right triangle (see the figure below).

Suppose you want to determine the distance d between the two points (x1, y1) and (x2, y2) in the plane. If the points lie on a horizontal line, then y1 = y2 and the distance between the points is | x2 - x1 |. If the points lie on a vertical line, then x1 = x2 and the distance between the points is | y2 - y1 |. If the two points do not lie on a horizontal or vertical line, they can be used to form a right triangle, as shown in the figure below.

The length of the vertical side of the triangle is | y2 - y1 | and the length of the horizontal side is | x2 - x1 |. By the Pythagorean Theorem, it follows that

Replacing | x2 - x1 | 2 and |y2 - y1 | 2 by the equivalent expressions (x2 - x1) 2 and (y2 - y1) 2 produces the following result.

Distance Formula

The distance d between the points (x1, y1) and (x2, y2) in the plane is given by

 

 

Example 1

Finding the Distance Between Two Points

Find the distance between the points (-2, 1) and (3, 4).

Solution

 

Example 2

Verifying a Right Triangle

Verify that the points (2, 1), (4, 0), and (5, 7) form the vertices of a right triangle.

Solution

The figure below shows the triangle formed by the three points.

The lengths of the three sides are as follows.

Because

d12 + d22 = 45 + 5 = 50 Sum of squares of sides

and

d32 = 50 Square of hypotenuse

you can apply the Pythagorean Theorem to conclude that the triangle is a right triangle.

 

Example 3

Using the Distance Formula

Find x such that the distance between (x, 3) and (2, -1) is 5.

Solution

Using the Distance Formula, you can write the following.

Distance Formula
25 = (x2 - 4x + 4) + 16 Square both sides.
0 = x2 - 4x - 5 Write in standard form.
0 = (x - 5)(x + 1) Factor.

Therefore, x = 5 or x = -1, and you can conclude that there are two solutions. That is, each of the points (5, 3) and (-1, 3) lies five units from the point as shown in the following figure.

The coordinates of the midpoint of the line segment joining two points can be found by “averaging” the x-coordinates of the two points and “averaging” the y-coordinates of the two points. That is, the midpoint of the line segment joining the points (x1, y1) and (x2, y2) in the plane is

For instance, the midpoint of the line segment joining the points (-5, -3) and (9, 3) is

as shown in the figure below

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