Coordinate geometry connects algebra and geometry on the SAT. If you know a handful of formulas and when to use them, these questions become quick wins.
The Distance Formula
The distance between two points and :
This is really just the Pythagorean theorem in disguise.
Example 1: Find the distance between and .
The Midpoint Formula
The midpoint of a segment connecting and :
Just average the x's and average the y's.
Example 2: Find the midpoint of and .
Equations of Lines
You should be comfortable with three forms:
| Form | Equation | Best For |
|---|---|---|
| Slope-intercept | Quick graphing, identifying slope | |
| Point-slope | Writing equations from a point + slope | |
| Standard | Some SAT answer choices |
Example 3: Write the equation of a line through with slope .
Finding Where Lines Intersect
Example 4: Where do and intersect?
Set them equal:
Intersection:
Horizontal and Vertical Lines
- Horizontal line through : (slope = 0)
- Vertical line through : (slope is undefined)
SAT Pro Tip: Sketch It
When a coordinate geometry question feels confusing, draw a quick sketch. Even a rough one helps you eliminate wrong answers.
Practice Problems
Problem 1: What is the distance between and ?
Solution
Problem 2: If is the midpoint of segment where , find .
Solution
Problem 3: A line passes through and . What is the equation in slope-intercept form?
Solution
Key Takeaways
- Distance formula = Pythagorean theorem on the coordinate plane
- Midpoint = average the coordinates
- Know all three forms of a linear equation and when each is useful
- To find intersections, set equations equal to each other
- When stuck, sketch the graph — even a rough one helps
Want more practice? Try NovaMaths — free SAT & ACT math prep with 750+ exercises and an AI tutor.