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 (x1, y1) and (x2, y2), then the distance d between them can be found by the Pythagorean Theorem:
Find the length of segment CD if the coordinates of C are (–4, 7) and the coordinates of D are (2, 3).
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 (x1, y1) and (x2, y2), 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.
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.
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.
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.
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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