The distance between two points is √((x₂ − x₁)² + (y₂ − y₁)²).
Step 1: Let's Learn
Read it, or press Listen and follow the words.
It is Pythagoras
The horizontal and vertical gaps are the legs of a right triangle and the distance is its hypotenuse. The formula is not new.
Signs do not matter here
The differences are squared, so subtracting in either order gives the same distance.
Midpoint
The midpoint is ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2). It is simply the average of each coordinate.
Do not mix them up
Distance subtracts and midpoint adds. Swapping the two is the most frequent error in this topic.
A quick check
The midpoint should look halfway between the points when sketched. If it does not, the arithmetic is wrong.
Distance is Pythagoras
The gap between two points is the hypotenuse of a right triangle with horizontal and vertical legs, giving √((x₂−x₁)² + (y₂−y₁)²). Sketching the triangle makes the formula derivable rather than memorised.
Order does not matter
The differences are squared, so the sign vanishes. Unlike the slope formula, the distance formula is forgiving about which point you subtract from which.
Midpoint is an average
((x₁+x₂)/2, (y₁+y₂)/2). Averaging each coordinate separately is exactly what halfway means in each direction, which is why the formula looks the way it does.
What they enable
Together they let you test whether sides are equal, find centres, and verify geometric claims numerically. They are the basic tools of coordinate proof.
Step 2: Try It Yourself
Tap and try it out.
The hypotenuse is not given. It comes from 3² + 4² = 25, whose square root is 5.
Step 3: Watch an Example
One step at a time.
Watch Rosa Measure a Segment
Rosa needs the distance and midpoint between (1, 2) and (4, 6).
- Step 1
The horizontal gap is 4 − 1 = 3 and the vertical gap is 6 − 2 = 4.
Step 4: Your Turn
Practice makes it stick.
The Distance
Problem 1 of 2
From (1, 2) to (4, 6). What is the distance?
The Middle
Problem 2 of 2
From (2, 4) to (8, 10). What is the x-coordinate of the midpoint?
Measure It
1 of 8
From (0, 0) to (3, 4). Distance?
2 of 8
From (0, 0) to (6, 8). Distance?
3 of 8
From (1, 1) to (1, 9). Distance?
4 of 8
From (2, 6) to (10, 6). Midpoint x-coordinate?
5 of 8
From (3, 1) to (3, 11). Midpoint y-coordinate?
6 of 8
From (−2, 0) to (4, 0). Distance?
7 of 8
Sort each operation by which formula uses it.
Tap something to move it.
- Empty
- Empty
8 of 8
From (0, 0) to (5, 12). Distance?
Step 5: Quick Check
Show what you know.
Question 1 of 2
From (0, 0) to (8, 6). What is the distance?
Question 2 of 2
Where does the distance formula come from?
What You Learned
- Distance is √((x₂ − x₁)² + (y₂ − y₁)²), which is Pythagoras.
- The midpoint averages each coordinate.
- Distance subtracts; midpoint adds.