doing math

courses › Geometry

Pythagoras & coordinates

level 23 course

Right triangles, distances, and midpoints.

Pen and paper is fine · no calculator needed why?

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Builds on: Exponents & roots (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

In a right triangle, the longest side, opposite the right angle, is the hypotenuse. The other two are the legs. Pythagoras' theorem ties them together: square the legs and add, and you get the hypotenuse squared. For legs 3 and 4: 9 + 16 = 25, so the hypotenuse is 5.

On a grid, the straight line between two points is the hypotenuse of a right triangle whose legs are the x-gap and the y-gap. That gives the distance. The midpoint is simpler: average the x-values and the y-values.

Techniques

Square, add or subtract, root

Finding one side of a right triangle from the other two.

  1. Square the two sides you know.
  2. Missing hypotenuse: add the squares. Missing leg: subtract the smaller square from the larger.
  3. Take the square root.
  4. Watch for 3, 4, 5 and 5, 12, 13, and those sets scaled up.
worked example

Example: A right triangle has hypotenuse 13 and one leg 5. How long is the other leg?

  1. 13 × 13 = 169, and 5 × 5 = 25.
  2. 169 − 25 = 144.
  3. 12 × 12 = 144, so the other leg is 12.

Answer: 12

Distance on a grid

The distance between two points given as (x, y).

  1. Find the gap in the x-values and the gap in the y-values. Ignore any minus sign.
  2. Use those gaps as the legs of a right triangle: square, add, take the root.
  3. If one gap is 0, the distance is the other gap.
worked example

Example: What is the distance between (−2, 1) and (4, 9)?

  1. x gap: 4 − (−2) = 6. y gap: 9 − 1 = 8.
  2. 6 × 6 + 8 × 8 = 100.
  3. 10 × 10 = 100, so the distance is 10.

Answer: 10

Average for the midpoint

The point halfway between two points.

  1. Add the two x-values and halve. That is the midpoint's x.
  2. Do the same with the y-values.
  3. Write the pair as x first, then y.
worked example

Example: What is the midpoint of (−6, 5) and (2, −1)? Give its x-coordinate.

  1. x: (−6 + 2) ÷ 2 = −2.
  2. y: (5 + (−1)) ÷ 2 = 2, so the midpoint is (−2, 2).

Answer: −2

Tips by skill

  • TipHypotenuse: Square both legs, add, then take the square root. Look for multiples of 3, 4, 5 or 5, 12, 13.
  • TipMidpoint: Average the two x-values, then the two y-values. Enter x first.
  • TipMissing leg: Square the hypotenuse, take away the square of the known leg, then take the square root.
  • TipDistance between points: Find the x-gap and the y-gap. Square them, add, and take the square root.

Watch out for

  • Adding the legs instead of their squares. Legs 9 and 12 give a hypotenuse of 15, not 21.
  • Adding the squares when the hypotenuse is given. The hypotenuse is the longest side, so a missing leg needs subtraction.
  • Forgetting the square root at the end.
  • Taking half the difference for a midpoint. Add the coordinates, then halve.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

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