Coordinate geometry is a fundamental concept in GMAT's quantitative reasoning section. It allows us to represent geometric shapes and relationships using algebraic equations and coordinates. Mastering this topic is crucial for solving problems involving lines, distances, areas, and more.

The Cartesian coordinate system, named after René Descartes, is the foundation of coordinate geometry. It uses two perpendicular number lines, the x-axis (horizontal) and the y-axis (vertical), to define the position of any point in a plane. The intersection of these axes is called the origin (0,0). A point is represented by an ordered pair (x, y), where 'x' is the horizontal distance from the origin and 'y' is the vertical distance.

The midpoint of a line segment is the point exactly halfway between its two endpoints. If the endpoints are $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint $(x_m, y_m)$ is found by averaging the x-coordinates and averaging the y-coordinates: $x_m = \frac{x_1 + x_2}{2}$ and $y_m = \frac{y_1 + y_2}{2}$.

The slope of a line measures its steepness and direction. It's defined as the 'rise over run' – the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line. For points $(x_1, y_1)$ and $(x_2, y_2)$, the slope 'm' is: $m = \frac{y_2 - y_1}{x_2 - x_1}$.

There are several forms to represent the equation of a line, each useful in different contexts:

• Slope-Intercept Form: $y = mx + b$, where 'm' is the slope and 'b' is the y-intercept (the point where the line crosses the y-axis).
• Point-Slope Form: $y - y_1 = m(x - x_1)$, where 'm' is the slope and $(x_1, y_1)$ is a point on the line.
• Standard Form: $Ax + By = C$, where A, B, and C are constants.

In GMAT Data Sufficiency questions, you'll often encounter coordinate geometry problems. The key is to determine if the given statements provide enough information to uniquely determine the answer. For example, to find the equation of a line, you typically need two pieces of information: either two points, or one point and the slope. A single point is not enough, nor is just knowing the slope without a point.

Consistent practice is vital. Work through a variety of GMAT-style problems, focusing on identifying the core geometric concepts being tested. For DS questions, always analyze each statement independently first, then consider them together. Sketching diagrams, even rough ones, can be incredibly helpful in visualizing the problem.