Coordinate geometry
Distance between points, midpoints and the section formula.
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Distance on a grid is Pythagoras: across and up are the two short sides.
Look: 6 across and 5 up. Straight distance = √(36 + 25) = √61 ≈ 7.8.
✏️ Your turn!
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1Distance from (0, 0) to (3, 4)?
2Middle of (2, 8) and (6, −2)?
3Distance from (0, 0) to (5, 12)?
! Careful!
✗Adding the coordinates.
✓Subtract, square, add, then √.
✗Forgetting the square root.
✓The last step is √.
See the whole lesson on one page
Distance on a grid is Pythagoras: across and up are the two short sides.
Find how far across and how far up. Square both, add, then take the square root. The midpoint is just the average of the two points.
Think of it like thisWalking across 6 blocks and up 5 is a long way. A bird flies straight: that straight line is the distance.
Solved together
How far is P(−2, 3) from Q(4, −5)? Where is the middle?
Across: from −2 to 4 is 6. Down: from 3 to −5 is 8.
Pythagoras.
√(6² + 8²) = √100 = 10Middle: average each coordinate.
((−2 + 4) ÷ 2, (3 − 5) ÷ 2) = (1, −1)
AnswerDistance 10. Middle (1, −1).
For older learners and grown-ups: a little more
Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]. Section formula (ratio m : n): ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).
Now practise it
Kanakku makes new questions on coordinate geometry every time. It shows the working and gives a hint when you are stuck.