How to Find Distance of Two Points? (+FREE Worksheet!)
The distance formula lets you calculate the exact length of any line segment on a coordinate plane using only the coordinates of its two endpoints. It comes directly from the Pythagorean theorem, so understanding one makes the other intuitive. Mastering this formula is a key skill for coordinate geometry throughout Algebra 1 and geometry.
What Is the Distance Formula?
For two points \(\color{blue}{(x_{1}, y_{1})}\) and \(\color{blue}{(x_{2}, y_{2})}\), the distance \(\color{blue}{d}\) between them is:
d = √((x2 − x1)2 + (y2 − y1)2)
You are finding the hypotenuse of a right triangle whose legs are the horizontal and vertical distances between the two points.
How to Use the Distance Formula
Step 1: Label the coordinates
Identify \(\color{blue}{(x_{1}, y_{1})}\) and \(\color{blue}{(x_{2}, y_{2})}\).
Step 2: Subtract x-coordinates and y-coordinates
Compute \(\color{blue}{x_{2} – x_{1}}\) and \(\color{blue}{y_{2} – y_{1}}\).
Step 3: Square each difference
\(\color{blue}{(x_{2} – x_{1})^{2}}\) and \(\color{blue}{(y_{2} – y_{1})^{2}}\). Note: squaring removes negative signs.
Step 4: Add and take the square root
Add the two squared differences and take the square root. Simplify if the result is a perfect square.
- Distance from \(\color{blue}{(0, 0)}\) to \(\color{blue}{(3, 4)}\): \(\color{blue}{d = \sqrt{(9 + 16)} = \sqrt{25} = 5}\)
Step-by-Step Summary
- Write down both endpoints and label them.
- Find \(\color{blue}{(x_{2} – x_{1})}\) and \(\color{blue}{(y_{2} – y_{1})}\).
- Square both differences.
- Add the squares.
- Take the positive square root of the sum.
- Simplify if possible.
Watch: Distance Formula (Video Lesson)
Khan Academy derives the distance formula from the Pythagorean theorem and works through examples:
Finding Distance of Two Points – Worked Examples
Example 1: Find the distance between \(\color{blue}{(0, 0)}\) and \(\color{blue}{(3, 4)}\).
\(\color{blue}{d = \sqrt{((3 – 0)}^{2} + (4 – 0)^{2}) = \sqrt{(9 + 16)} = \sqrt{25} = 5}\)
Answer: \(\color{blue}{5}\)
Example 2: Find the distance between \(\color{blue}{(1, 2)}\) and \(\color{blue}{(4, 6)}\).
\(\color{blue}{d = \sqrt{((4 – 1)}^{2} + (6 – 2)^{2}) = \sqrt{(9 + 16)} = \sqrt{25} = 5}\)
Answer: \(\color{blue}{5}\)
Example 3: Find the distance between \(\color{blue}{(-1, 3)}\) and \(\color{blue}{(2, 7)}\).
\(\color{blue}{d = \sqrt{((2 – (-1)})^{2} + (7 – 3)^{2}) = \sqrt{(9 + 16)} = \sqrt{25} = 5}\)
Answer: \(\color{blue}{5}\)
Example 4: Find the distance between \(\color{blue}{(-3, -4)}\) and \(\color{blue}{(3, 4)}\).
\(\color{blue}{d = \sqrt{((3 – (-3)})^{2} + (4 – (-4))^{2}) = \sqrt{(36 + 64)} = \sqrt{100} = 10}\)
Answer: \(\color{blue}{10}\)
More Practice: Distance Formula Video
This video works through finding the distance between two coordinate points with full step-by-step solutions:
Exercises for Finding the Distance of Two Points
Find the distance between each pair of points. Simplify all square roots.
- \(\color{blue}{(0, 0)}\) and \(\color{blue}{(5, 12)}\)
- \(\color{blue}{(2, 3)}\) and \(\color{blue}{(5, 7)}\)
- \(\color{blue}{(-1, -1)}\) and \(\color{blue}{(2, 3)}\)
- \(\color{blue}{(0, 4)}\) and \(\color{blue}{(3, 0)}\)
- \(\color{blue}{(-2, 1)}\) and \(\color{blue}{(2, 4)}\)
- \(\color{blue}{(1, 1)}\) and \(\color{blue}{(7, 9)}\)
Answers
- \(\color{blue}{13}\)
- \(\color{blue}{5}\)
- \(\color{blue}{5}\)
- \(\color{blue}{5}\)
- \(\color{blue}{5}\)
- \(\color{blue}{10}\)
Frequently Asked Questions
Where does the distance formula come from?
It comes from the Pythagorean theorem. The horizontal distance \(\color{blue}{|x_{2} – x_{1}|}\) and vertical distance \(\color{blue}{|y_{2} – y_{1}|}\) form the legs of a right triangle, and the segment between the two points is the hypotenuse: \(\color{blue}{c = \sqrt{(a^{2} + b^{2})}}\).
Can the distance be negative?
No. Distance is always a non-negative value. Even if the differences \(\color{blue}{(x_{2} – x_{1})}\) or \(\color{blue}{(y_{2} – y_{1})}\) are negative, squaring them makes them positive before you take the square root.
What is the difference between the distance formula and the midpoint formula?
The midpoint formula finds the coordinates of the center of a segment; the distance formula finds its length. Both use the same two endpoints.
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