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Coordinates in the head

Which quadrant, how far apart, the midpoint, the slope, the line through two points, the distance to a line: coordinate geometry, exact, without paper.

  • Geometry and trigonometry
  • Untimed
  • Endless levels

How it plays

Points on a grid, given as (x, y). Say which quadrant one is in, how far apart two are (Pythagoras), their midpoint (average each coordinate), the slope between them (rise over run), the line through them as y = mx + c, or how far a point is from a line.

Levels add the harder kinds in that order, and the coordinates spread wider as you climb.

Every figure is chosen so the answer is exact: distances come from Pythagorean triples, lines have whole slopes and intercepts. The wrong options are the real slips: the steps added instead of Pythagoras, half the difference for the midpoint, run over rise, the intercept's sign flipped.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This makes these coordinate-geometry calculations quicker and surer. It claims nothing wider.

Learn

How to get better at coordinates in the head: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Name the kind of question first — quadrant, distance, midpoint, slope, line, or distance to a line — and the formula that goes with it.
  2. Take the differences, second point minus first, keeping the signs: Δx = x₂ − x₁ and Δy = y₂ − y₁. Distance, slope and the line all start from them.
  3. Expect a tidy answer. Distances are whole, slopes are fractions in lowest terms, and midpoints are whole or end in .5. An untidy result means a slip.
  4. Check with one point before you commit: put a point into the line you chose, or see that the midpoint sits between the two x-values and the two y-values.

Techniques and tricks

  • Quadrant by the signs

    Read the signs of x and y and nothing else: (+, +) is I, (−, +) is II, (−, −) is III, (+, −) is IV — anticlockwise from the top right.

  • Spot the triple

    Every distance here is a Pythagorean triple: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29 and the multiples 6-8-10, 9-12-15, 12-16-20. Find the two differences and name the triple instead of squaring.

    e.g. (2, −3) to (−10, 2): across 12, up 5 — the 5-12-13 triangle, so 13.

  • Take out a common factor

    When both differences share a factor, divide it out, find the triple, and multiply back.

    e.g. Across 18, up 24: that is 6 × (3, 4), so the distance is 6 × 5 = 30.

  • Midpoint is an average

    Add the two x-values and halve; add the two y-values and halve. Watch the signs when one is negative.

    e.g. (−7, 3) and (4, −8): x = −3 ÷ 2 = −1.5, y = −5 ÷ 2 = −2.5, so (−1.5, −2.5).

  • Rise over run, same order

    Slope is Δy ÷ Δx with both differences taken in the same order, then cut to lowest terms. A slope going down from left to right is negative.

    e.g. (1, 2) to (4, 11): 9 ÷ 3 = 3. (−1, 4) to (3, −2): −6 ÷ 4 = −3/2.

  • Slope first, then c = y − mx

    For the line, find the slope, then put either point into c = y − mx. If one point has x = 0, its y is c already.

    e.g. (1, 5) and (3, 9): m = 4 ÷ 2 = 2, c = 5 − 2 × 1 = 3, so y = 2x + 3. Check (3, 9): 6 + 3 = 9.

  • The root under a distance to a line

    Distance = |ax₀ + by₀ + c| ÷ √(a² + b²). Here a and b always come from a triple, so √(a² + b²) is a whole number: 5, 10, 13, 17 or 15.

    e.g. (2, 1) from 3x + 4y − 5 = 0: |6 + 4 − 5| ÷ 5 = 1.

As the levels climb

  • It starts with quadrants and distances. Level 1 adds midpoints and slopes, level 3 the line through two points, and level 5 the distance from a point to a line. The newest kinds come up more often for a while.
  • After that the coordinates spread wider: up to ±6 at first, ±10 at level 2, ±14 at level 4, then 4 more each level until they reach ±86. From level 6, distances can use a triple doubled, such as 10-24-26.
  • The game is untimed unless you set “Time pressure”. With it on, a quadrant gets 8 s, midpoints and slopes 15 s, a distance 20 s, a line 30 s and a distance to a line 40 s, a little more as the numbers grow.

Traps to avoid

  • Adding the steps for a distance: across 5 and up 12 answered 17. That is the walk along the grid; the straight line is 13.
  • Forgetting the square root: 169 offered for 13. The sum of the squares is the distance squared.
  • Half the difference for the midpoint, or the coordinates swapped: (−2.5, −1.5) for (−1.5, −2.5). Half the difference says how far the midpoint is from one end, not where it is.
  • Run over rise, or one difference taken the other way round: 1/3 or −3 for a slope of 3. Write Δy on top, and take both differences from the same point.
  • The intercept with its sign flipped, y = 2x − 3 for y = 2x + 3, or a point’s y-value used as c. Check the line against both points.

How to practise

  • In “Kinds of question”, leave one kind on to drill it — Distance to a line has the most steps, and is worth practising alone.
  • Set “Answer by” to “Typing the answer”. A line is typed as y = mx + c with a plain minus — y = 2x - 1 — a slope of 1 as just x, and no + 0.
  • Once you are sure, turn “Time pressure” from “Off — untimed” to “Relaxed — 1.5× time”, then Normal.

Levels

It is untimed unless you turn the clock on. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More geometry and trigonometry

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