Key takeaway: The distance formula gives the straight-line length between two points on a coordinate plane: square the horizontal change, square the vertical change, add, then take the square root.
Grade 88.G.B.8Geometry and Measurement
Definition of Distance Formula
The distance formula finds how far apart two points are on a coordinate plane. It is the Pythagorean theorem in disguise: draw a horizontal segment and a vertical segment from the points, and those become the two legs of a right triangle whose hypotenuse is the distance you want.
Subtract the x-coordinates for one leg and the y-coordinates for the other. Which point comes first never matters, since squaring erases any minus sign. Used on a map grid, the distance formula measures a bee-line.
Formula and notation
Let \(\Delta x\) be the run and \(\Delta y\) the rise between the two points, found by subtracting matching coordinates. The distance \(d\) is
\[d = \sqrt{(\Delta x)^{2} + (\Delta y)^{2}}\]
With the points written \((x_1, y_1)\) and \((x_2, y_2)\), the run is \(x_2 – x_1\). The answer uses the unit of the grid.
Key facts
- The distance between two points is never negative, and it is zero only for the same point.
- Swapping the two points gives the same distance.
- Points on a horizontal or vertical line need only a subtraction.
- The result is often a square root, so it may need a decimal approximation.
Distance Formula example
On a trail map drawn on a coordinate grid (1 unit = 1 mile), a cabin sits at \((-6, 4)\) and a lookout tower at \((9, -4)\). How far apart are they in a straight line?
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Horizontal change: \(9 – (-6) = 15\). Vertical change: \(-4 – 4 = -8\).
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Square each change: \(15^{2} = 225\) and \((-8)^{2} = 64\).
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Add and take the root: \(d = \sqrt{225 + 64} = \sqrt{289} = 17\).
The cabin and the tower are 17 miles apart.
Common mistake
Watch out
Hugo computes \(d = \sqrt{(9 – 6)^{2} + (-4 – 4)^{2}} = \sqrt{9 + 64} \approx 8.5\) for the same trail points.
Why it is wrong
He dropped the negative sign of \(-6\): the horizontal change is \(9 – (-6) = 15\), not 3. Put parentheses around negative coordinates before subtracting. With the right sign, \(\sqrt{225 + 64} = 17\).
Related terms and lessons
- Grade 8 · Distance in the Coordinate PlaneLessonWorksheetQuiz
- Geometry · Coordinate Geometry ProofsLessonWorksheetQuiz
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