Distance Between Two Points

2 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

Join two points with a straight line. Then draw one horizontal line and one vertical line to make a corner.

You now have a right triangle, and your line is its longest side. The two short sides are $|x_2 - x_1|$ and $|y_2 - y_1|$ . So the Pythagorean theorem gives you the distance.

Rule

For any two points $A(x_1, y_1)$ and $B(x_2, y_2)$ :

|AB| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Because the differences are squared, the subtraction order does not matter: $(x_1 - x_2)^2 = (x_2 - x_1)^2$ .

How it is used

Three standard patterns:

Direct computation. Plug both points in and simplify - leave answers in exact form like $5\sqrt{2}$ .
Distance to the origin. With $O(0, 0)$ the formula collapses to $\sqrt{x^2 + y^2}$ .
Unknown coordinate. If you are given the distance, square both sides and solve. Remember the square root gives a plus and a minus, so there are usually two answers.

Worked example

Find the distance between $A(-2, 4)$ and $B(3, -1)$ . Differences: $x_2 - x_1 = 3 - (-2) = 5$ and $y_2 - y_1 = -1 - 4 = -5$ . $|AB| = \sqrt{5^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50}$ Simplify: $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$ units.

Explore

Drag either point. The dashed sides are the differences in $x$ and in $y$ . The blue side is the distance. Look for the cases where the distance is a whole number. Those are the 3-4-5 and 5-12-13 triangles.

Distance between two points

Try this. Place A at (0, 0) and B at (3, 4): the distance is 5. Swap the points; the distance stays the same.

Point A · x-2Point A · y3Point B · x3Point B · y1

Distance = √((5)² + (-2)²) = √29 ≈ 5.385

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

What is the distance from

(1,2)

(4,6)

$7$

$25$

$5$

Hint 1 · Find a starting point

Draw the right triangle joining the points.

Hint 2 · Take the next step

The changes are 3 horizontally and 4 vertically; square them, add, then take a square root.

Show the reasoning

Answer: $5$

$d=\sqrt{3^2+4^2}=5$ . Adding the changes gives a path along the axes, not the straight-line distance.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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