The distance between two points is Pythagoras applied to the horizontal and vertical gaps.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The formula
d = √((x₂ − x₁)² + (y₂ − y₁)²). It is not new; it is the theorem in coordinates.
The midpoint
Average the x-coordinates and average the y-coordinates.
A circle is a distance condition
(x − h)² + (y − k)² = r² says every point is r from the centre (h, k).
Distance is Pythagoras
The distance between two points is √((x₂−x₁)² + (y₂−y₁)²), the hypotenuse of a right triangle with horizontal and vertical legs. Sketching that triangle makes the formula something you derive rather than recall.
Midpoint is an average
The midpoint is ((x₁+x₂)/2, (y₁+y₂)/2) — the average of the coordinates. Averaging each coordinate separately is exactly what "halfway" means in each direction.
The circle equation is the distance formula
(x − h)² + (y − k)² = r² says every point on the circle is distance r from the centre (h, k). It is the distance formula with both sides squared, which is why the equation looks the way it does.
Completing the square finds the centre
An equation like x² + y² − 6x + 4y = 12 hides its centre until you complete the square in both variables. The technique from Algebra 1 is the tool that converts a general equation into recognisable circle form.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Sam Find a Distance
From (1, 2) to (7, 10).
- Step 1
The horizontal gap is 7 − 1 = 6.
Step 4: Your Turn
Practice makes it stick.
The Distance
Problem 1 of 2
From (0, 0) to (3, 4). What is the distance?
The Midpoint
Problem 2 of 2
From (2, 4) to (8, 10). What is the x-coordinate of the midpoint?
Coordinates and Distance
1 of 4
From (0, 0) to (6, 8). Distance?
2 of 4
From (1, 1) to (4, 5). Distance?
3 of 4
From (3, 7) to (9, 11). Midpoint x-coordinate?
4 of 4
x² + y² = 49. What is the radius?
Step 5: Quick Check
Show what you know.
Question 1 of 2
From (2, 3) to (10, 9). Distance?
Question 2 of 2
What is the distance formula really?
What You Learned
- Distance is Pythagoras applied to the coordinate gaps.
- The midpoint averages each coordinate.
- A circle equation states that every point is a fixed distance from the centre.