Distance between two points
On a grid, the straight-line distance between two points is the hypotenuse of a right triangle. The horizontal leg is the change in x and the vertical leg is the change in y. By the Pythagorean theorem:
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
This is the distance formula.
Example: A(1, 2) and B(7, 10)
- Δx = 6, Δy = 8
- d = √(36 + 64) = √100 = 10
Example: P(−3, 4) and Q(2, −8)
- Δx = 5, Δy = −12
- d = √(25 + 144) = √169 = 13
The order of the points doesn’t matter: squaring removes negatives.
Midpoint of a segment
The midpoint is halfway between the endpoints of a line segment. Average the coordinates:
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
This is the midpoint formula.
Example: A(2, 5) and B(8, −1)
M = ((2 + 8)/2, (5 + (−1))/2) = (5, 2)
Finding an endpoint
If you know the midpoint M and one endpoint A, find B by going "the same again".
Example: M(4, 3), A(1, −2)
- From A to M: +3 in x, +5 in y
- B = (4 + 3, 3 + 5) = (7, 8)
Applications
Classifying triangles: Find all three side lengths.
Triangle with vertices (0, 0), (4, 0), (2, 5):
- sides 4, √29, √29 → isosceles
Checking a right angle: If a² + b² = c² for the side lengths, it’s a right triangle.
Perpendicular bisector: The midpoint gives a point on it; the negative reciprocal slope gives its direction.
Circle centre: The midpoint of a diameter is the centre of the circle.
Real-world: Map coordinates → straight-line distance between two towns.
Worked examples
Example 1: Distance from (0, 0) to (5, 12)? √(25 + 144) = 13.
Example 2: Midpoint of (−4, 6) and (10, 2)? (3, 4).
Example 3: Distance from (3, 1) to (3, 9)? Δx = 0 → 8.
Common mistakes
- Forgetting to square or to take the square root.
- Subtracting coordinates in mixed order for the midpoint (it uses adding).
- Treating a negative difference incorrectly: (−3 − 2)² = 25, not −25.
Try it yourself
- Distance between (2, 3) and (5, 7)
- Distance between (−1, −1) and (4, 11)
- Midpoint of (0, 8) and (6, −2)
- M(1, 1) is the midpoint of AB with A(−3, 5). Find B.
- Is the triangle (0, 0), (3, 0), (0, 4) a right triangle?
Answers: 1) 5 2) 13 3) (3, 3) 4) (5, −3) 5) yes (3, 4, 5)
Quick recap
- Distance formula comes from the Pythagorean theorem.
- Midpoint formula averages the coordinates.
- Use both to classify shapes and find centres.