Find a Distance Between Coordinates
Build a right triangle from coordinate differences.
Supplied materials
- Paper or a note app; a calculator is optional.
- Supplied points: P = (−2,1), Q = (4,9). Coordinates use the same length unit.
Learn the idea
A horizontal and a vertical move between two points form perpendicular legs. Their lengths are the magnitudes of the coordinate differences. From P to Q the horizontal change is 4 − (−2) = 6 and the vertical change is 9 − 1 = 8.
The straight-line distance is the hypotenuse, so d = √[(x₂ − x₁)² + (y₂ − y₁)²]. Squaring removes the sign of each difference. Adding the absolute differences gives a route along perpendicular edges, not the direct distance.
- Mark P and Q and draw a horizontal leg followed by a vertical leg.
- Calculate the leg lengths 6 and 8, keeping coordinate order consistent.
- Apply the Pythagorean theorem and compare the diagonal with the 14-unit edge route.
See the method in action
d = √[(4 + 2)² + (9 − 1)²] = √(36 + 64) = 10 units. Reversing the differences gives (−6)² + (−8)², the same squared distance.
Check your reasoning
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See the answer and explanation
10 units
The horizontal difference is 8 and the vertical difference is 6. d = √(8² + 6²) = √100 = 10 units.
Take it further
Move the second point to (4,6), keeping the first at (1,2). The legs become 3 and 4, so d = √(9 + 16) = 5 units.
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