Lines and Distance
Lines and distance are fundamental to coordinate geometry,
not to mention to the Math IC test. Even the most complicated coordinate
geometry question uses the concepts covered in the next few sections.
Measuring distance in the coordinate plane is made possible
thanks to the Pythagorean theorem. If you are given two points,
their distance from each other is given by the following formula:
The diagram below shows how the Pythagorean theorem plays
a role in the formula. The distance between two points can be represented
by the hypotenuse of a right triangle whose legs are lengths (x2 – x1)
and (y2 – y1).
To calculate the distance from (4, –3) to (–3, 8), plug
the coordinates into the formula:
The distance between the points is
, which equals approximately 13.04.
You can double-check this answer by plugging it back into the Pythgorean
The midpoint between two points in the coordinate plane
can be calculated using a formula. If the endpoints of a line segment
are (x1, y1)
and (x2, y2),
then the midpoint of the line segment is:
In other words, the x- and y-coordinates
of the midpoint are the averages of the x- and y-coordinates
of the endpoints.
Here’s a practice question:
is the midpoint of the line segment whose endpoints are (6, 0) and
To solve, all you need to do is plug the points given
into the midpoint formula. x1 = 6, y1 = 0, x2 =
3, and y2 = 7: