Pythagoras & coordinates
Right triangles, distances, and midpoints.
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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
- Square the two sides you know.
- Missing hypotenuse: add the squares. Missing leg: subtract the smaller square from the larger.
- Take the square root.
- Watch for 3, 4, 5 and 5, 12, 13, and those sets scaled up.
worked example
A right triangle has hypotenuse 13 and one leg 5. How long is the other leg?
- 13 × 13 = 169, and 5 × 5 = 25.
- 169 − 25 = 144.
- 12 × 12 = 144, so the other leg is 12.
Answer: 12
Distance on a grid
- Find the gap in the x-values and the gap in the y-values. Ignore any minus sign.
- Use those gaps as the legs of a right triangle: square, add, take the root.
- If one gap is 0, the distance is the other gap.
worked example
What is the distance between (−2, 1) and (4, 9)?
- x gap: 4 − (−2) = 6. y gap: 9 − 1 = 8.
- 6 × 6 + 8 × 8 = 100.
- 10 × 10 = 100, so the distance is 10.
Answer: 10
Average for the midpoint
- Add the two x-values and halve. That is the midpoint's x.
- Do the same with the y-values.
- Write the pair as x first, then y.
worked example
What is the midpoint of (−6, 5) and (2, −1)? Give its x-coordinate.
- x: (−6 + 2) ÷ 2 = −2.
- y: (5 + (−1)) ÷ 2 = 2, so the midpoint is (−2, 2).
Answer: −2
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.
practice
Hypotenuse
worked example
A right triangle has legs 42 and 40. How long is the hypotenuse?
Answer: 58
- √(42² + 40²) = √(1,764 + 1,600) = √3,364 = 58.
- It is the 20-21-29 triangle scaled by 2.
Midpoint
worked example
What is the midpoint of (−5, −4) and (−1, −8)? Enter x, y.
Answer: (-3, -6)
- Average each coordinate: (−5 + (−1)) ÷ 2 = −3 and (−4 + (−8)) ÷ 2 = −6.
- The midpoint is (−3, −6).
Missing leg
worked example
A right triangle has hypotenuse 51 and one leg 24. How long is the other leg?
Answer: 45
- √(51² − 24²) = √(2,601 − 576) = √2,025 = 45.
Distance between points
worked example
What is the distance between (3, −8) and (12, 4)?
Answer: 15
- The x-values differ by 9 and the y-values by 12.
- Distance = √(9² + 12²) = √(81 + 144) = √225 = 15.