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).
![](./articles_imgs/120/algebr10.gif)
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.
![](./articles_imgs/120/algebr11.gif)
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
![](./articles_imgs/120/algebr12.gif)
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
![](./articles_imgs/120/algebr13.gif)
Example 1
Finding the Distance Between Two Points
Find the distance between the points (-2, 1) and (3, 4).
Solution
![](./articles_imgs/120/algebr14.gif)
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.
![](./articles_imgs/120/algebr15.gif)
The lengths of
the three sides are as follows.
![](./articles_imgs/120/algebr16.gif)
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.
![](./articles_imgs/120/algebr17.gif) |
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.
![](./articles_imgs/120/algebr18.gif)
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
![](./articles_imgs/120/algebr19.gif)
For instance, the midpoint of the line segment joining the points (-5, -3) and
(9, 3)
is
![](./articles_imgs/120/algebr20.gif)
as shown in the figure below
![](./articles_imgs/120/algebr21.gif)
|