The distance formula gives the length of the straight line segment between two points on a coordinate plane. It is one of the most useful tools in geometry because it connects coordinates, graphing, and measurement. Instead of using a ruler, you can calculate an exact distance from the points' x and y values.
This matters in geometry, physics, mapping, computer graphics, and any situation involving position.
Key Facts
- Distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
- Horizontal change: Δx = x2 - x1
- Vertical change: Δy = y2 - y1
- The distance formula comes from the Pythagorean theorem: a^2 + b^2 = c^2
- For points A(x1, y1) and B(x2, y2), the distance d is the length of segment AB.
- Squaring Δx and Δy makes the distance nonnegative, so the order of the points does not change the answer.
Vocabulary
- Distance formula
- A formula used to find the straight-line distance between two points on a coordinate plane.
- Coordinate plane
- A flat grid formed by a horizontal x-axis and a vertical y-axis where points are located by ordered pairs.
- Ordered pair
- A pair of numbers written as (x, y) that gives the location of a point on the coordinate plane.
- Horizontal change
- The difference in x-values between two points, often written as Δx.
- Vertical change
- The difference in y-values between two points, often written as Δy.
Common Mistakes to Avoid
- Forgetting to square both coordinate differences is wrong because the formula depends on the legs of a right triangle, so both Δx and Δy must be squared.
- Subtracting coordinates in different orders is wrong because each difference must use the same point order, such as x2 - x1 and y2 - y1.
- Leaving the answer as d^2 instead of d is wrong because distance is the square root of the sum, not just the sum of the squares.
- Treating negative coordinate differences as negative distances is wrong because distance is always nonnegative and the squares remove sign direction.
Practice Questions
- 1 Find the distance between A(2, 3) and B(8, 11).
- 2 Find the length of the segment with endpoints P(-4, 5) and Q(6, -7).
- 3 Explain why the distance from A to B is the same as the distance from B to A, even though the coordinate differences may have opposite signs.