Distance and Midpoint Formulas
Two points on a grid, and the test wants to know how far apart they are or what's exactly between them. These two formulas turn that into quick arithmetic — and they show up in coordinate geometry and circle problems too.
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.
Distance is the hypotenuse of a right triangle whose legs are the horizontal and vertical gaps.
Description of the figure:
- The x-axis runs from negative 1 to 7 and the y-axis from negative 1 to 12, with grid lines every 1 unit.
- The line rises from left to right.
- It crosses the y-axis at (0, 2.5) and does not meet the x-axis within the graph.
- It also passes through (negative 1, 1) and (5, 10).
- Marked points: A at (1, 4), M at (3, 7), and B at (5, 10).
The midpoint M sits exactly halfway between endpoints A and B.
A quick mental check: distance uses subtraction and a square root; midpoint uses addition and division by 2. Mixing them up is the #1 error, so anchor on that.
What is the distance between (0, 0) and (3, 4)?
Worked examples
What is the distance between the points (1, 2) and (4, 6)?
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.)
Common pitfalls
After adding the squares you get a number like 25, and it's tempting to stop. But distance is √25 = 5. The unsquared sum is almost always a wrong-answer choice.
Distance subtracts, squares, adds, and roots. Midpoint adds and divides by 2. Anchor on 'midpoint = average' so you never divide in a distance problem.
When a coordinate is negative, like (−1, 6), subtraction becomes addition: 4 − (−1) = 5. Write the substitution out fully instead of doing it in your head.
When the problem gives the midpoint and one endpoint, you're solving for the other endpoint — set up midpoint = (known + unknown)/2 and solve, don't just average the two points you see.
Key takeaways
Distance:
d = √[(x₂−x₁)² + (y₂−y₁)²]— it's the Pythagorean theorem in disguise.Midpoint:
M = ((x₁+x₂)/2, (y₁+y₂)/2)— just average each coordinate.To find a missing endpoint, set up
midpoint = (known + unknown)/2and solve each coordinate separately.Always finish distance with the square root; the unsquared sum is a trap answer.
Substitute negative coordinates carefully — subtracting a negative becomes adding.
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.