Find the Distance Between Two Points: Step-by-Step Guide
The distance between two points is the length of the straight line that connects them. You can find it by making a right triangle. The horizontal difference and the vertical difference are the two legs, and the distance is the hypotenuse. This gives the distance formula: d = square root of ((x2 - x1)^2 + (y2 - y1)^2).
Math Racer skill guide · Geometry · Grade 10
How to solve it, step by step
- Label the two points. Call them (x1, y1) and (x2, y2). The order does not matter, because we will square the differences.\[(x_1, y_1),\;(x_2, y_2)\]
- Find the differences. Work out the horizontal difference, x2 - x1, and the vertical difference, y2 - y1.\[\Delta x = x_2 - x_1,\;\Delta y = y_2 - y_1\]
- Square both and add. Square each difference, then add the two results.\[(\Delta x)^2 + (\Delta y)^2\]
- Take the square root. The distance is the positive square root of the sum.\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
Worked example
Find the distance between (1, 2) and (4, 6).
Step 1: Call the points (x1, y1) = (1, 2) and (x2, y2) = (4, 6).
Step 2: The differences: x2 - x1 = 4 - 1 = 3, and y2 - y1 = 6 - 2 = 4.
Step 3: Square them: 3^2 = 9 and 4^2 = 16. Add: 9 + 16 = 25.
Step 4: Take the square root: d = the square root of 25 = 5.
Check: This is the famous 3-4-5 right triangle, so the distance is 5.
The distance is 5.
Tips and common mistakes
- The distance formula is the Pythagorean theorem in disguise: a^2 + b^2 = c^2.
- The order does not matter. (x2 - x1)^2 and (x1 - x2)^2 are always the same, because squaring removes the negative.
- Square first, then add. (a + b)^2 is not the same as a^2 + b^2.
- Distance is never negative. Use the positive square root.
- Simplify square roots when you can. The square root of 50 is 5 times the square root of 2.
- If two points have the same x, the distance is just the difference in y (as a positive number). If they have the same y, it is the difference in x.
- If the answer is not a whole number, you can leave it as a square root unless the problem asks for a decimal.