Coordinate Geometry on the SAT

Why the SAT Emphasizes Coordinate Geometry

Coordinate geometry blends algebra and geometry. The SAT often asks for distances, midpoints, and slopes between points, which means you need to move smoothly between formulas and the coordinate plane.

This lesson shows how to use the distance and midpoint formulas, how to set up calculations cleanly, and how to avoid sign errors when subtracting coordinates.

A Simple Definition Unlocks Coordinate Geometry

The distance formula comes from the Pythagorean Theorem, and the midpoint formula is the average of the two coordinates. Both are straightforward once you organize the numbers.

Always subtract in the same order when finding differences. Mixing the order in one place but not the other creates sign mistakes that are easy to miss.

Work Through Coordinate Geometry Step by Step

Find the distance between $(1, 2)$ and $(5, 5)$ using the distance formula step by step.

Use Desmos to Check Coordinate Geometry

Desmos can help verify distance by using the distance formula directly or by plotting points and measuring. It is a good check for arithmetic.

Algebra is faster for simple coordinates. Desmos is useful when coordinates are messy.

Desmos is faster for messy distances or midpoint calculations. Algebra is faster when the numbers are small and clean.

Expert move: Use Desmos for numeric checks (distance, midpoint, intercepts) when coordinates are given, but do not use it to replace geometry rules or angle chasing.

When to skip Desmos: For simple relationships (vertical angles, triangle sum), the algebra is faster.

• Desmos features used: numeric evaluation.
• Common mistake: switching one coordinate and not the other.

Practice Coordinate Geometry with SAT-Style Questions

Use distance and midpoint formulas.

Key Takeaways to Remember for Coordinate Geometry

• Distance formula uses squared differences in $x$ and $y$.
• Midpoint is the average of the coordinates.
• Desmos is helpful for checking arithmetic quickly.