Coordinate Geometry
Plotting points and measuring distances on a grid shows up across the Math section — and these questions are almost always free points once you memorize three formulas.
Watch & learn
Short walkthroughs on Coordinate Geometry — ours, plus curated picks. Watch one if a written explanation isn't clicking, or after to reinforce.
Description of the figure:
- The right angle is at the bottom-left vertex, between a horizontal leg and a vertical leg.
- The horizontal leg is labeled Δx = 3, the vertical leg is labeled Δy = 4, and the hypotenuse is labeled d = 5.
The distance formula is the Pythagorean theorem: horizontal and vertical gaps are the legs.
Description of the figure:
- The x-axis runs from negative 2 to 6 and the y-axis from negative 2 to 12, with grid lines every 1 unit.
- The line rises from left to right.
- It crosses the y-axis at (0, 1) and meets the x-axis at (negative 0.5, 0).
- It also passes through (negative 1, negative 1).
- Marked points: (1, 3) and (3, 7).
- A slope triangle is drawn from (1, 3) to (3, 7), labeled run = 2 and rise = 4.
Slope = rise/run. From (1,3) to (3,7): rise 4, run 2, slope = 2.
Check your understanding with a question from this topic:
What is the distance between (0, 0) and (3, 4)?
Worked examples
What is the distance between the points (0, 0) and (3, 4)?
The midpoint of segment AB is M(3, 7). If A = (1, 4), what are the coordinates of B?
The midpoint of segment PQ is M(5, −2). If P = (−1, 6), what is the sum of the coordinates of Q? (Enter your answer as a number.)
Common pitfalls
Students sometimes write √(Δx + Δy) instead of √(Δx² + Δy²). The squares must come BEFORE the addition under the radical, or you get the wrong distance.
When given the midpoint and one endpoint, the other endpoint is often OUTSIDE the range you'd expect. Don't just split the difference — set the average equal to the midpoint and solve algebraically.
Subtracting a negative flips to addition: 3 − (−1) = 4. Mishandling this in distance or slope calculations is the #1 source of wrong answers here.
Parallel lines have equal slopes; perpendicular lines have negative reciprocal slopes (product = −1). Flipping these gives a perfectly wrong-looking answer.
Key takeaways
Distance:
d = √[(x₂−x₁)² + (y₂−y₁)²]— it's the Pythagorean theorem on a grid.Midpoint: average the x's and average the y's; to find a missing endpoint, set the average equal to the midpoint and solve.
Slope = rise/run =
(y₂−y₁)/(x₂−x₁).Parallel slopes are equal; perpendicular slopes are negative reciprocals (multiply to −1).
Track negative signs carefully — they cause most errors in this topic.
Further reading
Skill check
10 questions on Coordinate Geometry, from our question bank. Score 70% or better across 10 answers to reach Silver and pass this step.