Math

Coordinate Geometry

2 min readEasy5-question drill

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.

Where this lesson fits

Coordinate Geometry

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Coordinate geometry is just doing geometry on the xy-plane — the grid where every point is written as (x, y). The first number tells you how far left/right (horizontal), the second how far up/down (vertical). Three tools handle almost every question.

1. Distance formula — how far apart two points are. For points (x₁, y₁) and (x₂, y₂):

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

This is just the Pythagorean theorem in disguise: the horizontal gap and the vertical gap are the two legs of a right triangle, and the distance is the hypotenuse.

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The distance formula is the Pythagorean theorem: horizontal and vertical gaps are the legs.

2. Midpoint formula — the exact center point of a segment. Average the x's and average the y's:

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

A common twist gives you the midpoint and ONE endpoint and asks for the other. Just plug in and solve. For example, if midpoint = (3, 7) and one endpoint is (1, 4), set (1 + x)/2 = 3 → x = 5.

3. Slope — the steepness of a line connecting two points:

slope = (y₂ − y₁)/(x₂ − x₁) = rise over run.

Two key rules:

Parallel lines have equal slopes.
Perpendicular lines have slopes that are negative reciprocals — multiply to −1. So a line with slope 2 is perpendicular to one with slope −1/2.

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Slope = rise/run. From (1,3) to (3,7): rise 4, run 2, slope = 2.

Tips that save time:

Order doesn't matter in distance or midpoint as long as you're consistent — (x₂ − x₁) squared equals (x₁ − x₂) squared.
Watch your negative signs. Subtracting a negative becomes addition: 3 − (−1) = 4.
For midpoint problems solving for an endpoint, remember the endpoint can be FARTHER out than the midpoint — don't expect a number between the two given values.

That's the whole toolkit. Most test questions are direct plug-ins; harder ones bury these formulas inside a word problem or a figure.

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Quick check

Check your understanding with a question from this topic:

What is the distance between (0, 0) and (3, 4)?

Worked examples

Example 1

What is the distance between the points (0, 0) and (3, 4)?

Example 2

The midpoint of segment AB is M(3, 7). If A = (1, 4), what are the coordinates of B?

Example 3

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

Forgetting to square inside the distance formula

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.

Averaging instead of solving for a missing endpoint

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.

Sign errors with negatives

Subtracting a negative flips to addition: 3 − (−1) = 4. Mishandling this in distance or slope calculations is the #1 source of wrong answers here.

Mixing up parallel and perpendicular slopes

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.

Watch & learn

Curated Khan Academy walkthroughs on Coordinate Geometry. They're complementary to this lesson — watch one if a written explanation isn't clicking, or after to reinforce.

Tracks your progress across lessons.

Try it yourself

5 practice questions on Coordinate Geometry, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Coordinate Geometry

You are here

→Linear Equations in 2 Variables →Lines, Angles, and Triangles →Circles