Coordinate Geometry
and Distance
Master the coordinate plane, the distance formula, and the midpoint formula: essential tools for GRE Quantitative success.
The Coordinate Plane
Every point in the plane is described by an ordered pair \((x, y)\). The horizontal axis is the \(x\)-axis, and the vertical axis is the \(y\)-axis. They intersect at the origin \((0, 0)\).
Quadrant Sign Rules
Why Coordinate Geometry Matters
Coordinate geometry converts geometric relationships into algebra. On the GRE, you will use it to find distances, midpoints, slopes, and to verify whether shapes have specific properties.
The Distance Formula
For two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\):
The Midpoint Formula
The midpoint \(M\) of a segment joining \(A(x_1, y_1)\) and \(B(x_2, y_2)\):
Distance vs. Midpoint
| Distance | Midpoint | |
|---|---|---|
| Returns | A number (length) | A point (coordinates) |
| Operation | Subtract, square, add, root | Add, then divide by 2 |
| Based on | Pythagorean Theorem | Arithmetic Mean |
| GRE Cue | "How far," "length of" | "Center," "halfway," "bisects" |
How to Apply the Formulas
Example 1: Distance Between Two Points
Find the distance between \(P(-3, 4)\) and \(Q(5, -2)\).
Example 2: Finding the Midpoint
Find the midpoint of the segment from \(A(-6, 2)\) to \(B(4, 8)\).
Example 3: Finding an Endpoint
The midpoint of segment \(\overline{AB}\) is \(M(3, 7)\). If \(A = (1, 4)\), find \(B\).
Special Cases: Horizontal and Vertical
Using Distance to Find Area
On the GRE, you may be asked for the area of a triangle or rectangle given coordinates. Use the distance formula to find side lengths, then apply the appropriate area formula.
Common Mistakes
Formula Summary
| Concept | Formula | Result Type |
|---|---|---|
| Distance | \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\) | A number |
| Midpoint | \(\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\) | A point |
| Horizontal distance | \(|x_2 - x_1|\) | A number |
| Vertical distance | \(|y_2 - y_1|\) | A number |
| Reverse midpoint | \(x_2 = 2M_x - x_1\) | A coordinate |
Question 1 of 5
What is the distance between the points \((2, -1)\) and \((-4, 7)\)?
Question 2 of 5
What is the midpoint of the segment joining \((-2, 6)\) and \((8, -4)\)?
Question 3 of 5
The midpoint of \(\overline{PQ}\) is \(M(4, 3)\). If \(P = (1, -2)\), what are the coordinates of \(Q\)?
Question 4 of 5
A triangle has vertices at \((0, 0)\), \((6, 0)\), and \((6, 8)\). What is the length of the hypotenuse?
Question 5 of 5
Point \(A\) is at \((-3, 5)\) and point \(B\) is at \((9, 5)\). What is the distance from \(A\) to \(B\)?