**Thie Distance and Midpoint Formulas**

**The Distance Formula**

Points *A* and *B* have coordinates (2, 3) and (5, 7). The distance from point *A* to point *B* is the length of segment *AB*. You can find that distance by drawing a right triangle, finding the lengths of its legs, and then using the Pythagorean Theorem:

Notice that the horizontal leg's length can be found by subtracting the *x*-coordinates of the points and the vertical leg's length can be found by subtracting the *y*-coordinates of the points. In general, if the points have coordinates (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}), then the distance *d* between them can be found by the Pythagorean Theorem:

or

Example:

Find the length of segment *CD* if the coordinates of *C* are (–4, 7) and the coordinates of *D* are (2, 3).

Solution:

**The Midpoint Formula**

On the number line, suppose point *A* has coordinate 17 and point *B* has coordinate 25. What is the coordinate of the midpoint of segment *AB*?

You can see from the diagram that the coordinate is 21. A shortcut (if you don't want to draw part of a number line) is to find the *average* of the coordinates:

In 2 dimensions, the coordinates of the midpoint of two points is the *average* of their coordinates. For example, if the coordinates of *A* are (2, 3) and the coordinates of *B* are (10, 11), then the coordinates of the midpoint *M* of segment *AB* are (6, 7).

In general, if the points have coordinates (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}), then the midpoint *M* has coordinates that are the *average* of these:

**An Equation of a Circle**

An equation of any shape graphed in the *xy*-coordinate plane is a statement about all the points on it, using *x* and *y* as the coordinates. If you have a circle of radius 5 centered at the origin, the Pythagorean Theorem tells us something that must be true of all points on the circle:

** **

In general, the equation of a circle of radius *r* centered at the origin is:

**The Rhombus Problems**

A rhombus is a parallelogram in which all four sides are equal. It is like a "slanted square." Suppose points *A*, *B*, *C*, and *D* have coordinates (0, 0), (3, 4), (8, 4), and (5, 0), respectively:

1. Show that *ABCD* is a parallelogram.

Solution:

Sides *AD* and *BC* are parallel because they are both horizontal. Sides *AB* and *CD* are parallel if they have the same slope:

2. Show that *ABCD* is a rhombus.

Solution:

We already know it is a parallelogram, so now we must show all sides have the same length. Since sides *AD* and *BC* are horizontal, their lengths are easy to find:

Now we use the distance formula to find the lengths of the other sides:

So all 4 sides have the same length.

3. Show that the diagonals of *ABCD* bisect each other.

Solution:

The easiest way to do this is to show the midpoints of segments *AC* and *BD* are the same point:

4. Show that the diagonals of *ABCD* are perpendicular.

Solution:

Two lines are perpendicular if their slopes are opposite reciprocals, so we find the slopes of segments *AC* and *BD*:

Since 1/2 and –2 are opposite reciprocals, the diagonals are perpendicular.

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