Derivation/Proof of Theorems: Coordinate Geometry

This section derives and explains the distance formula in two-dimensional Cartesian coordinates. The derivation uses the Pythagorean theorem and gives simple examples, special cases and brief applications so that the result is clear and useful for problem solving.

Theorem: Distance Formula

Statement:

f $P(x1,y1)P(x\_1,\; y\_1)$P(x1,y1) and $Q(x2,y2)Q(x\_2,\; y\_2)$Q(x2,y2) are two points in the Cartesian plane, then the distance between them is

Given: Two points P and Q with coordinates (x1,y1) and (x2,y2) respectively.

To prove: 

Proof

Consider points P(x1,y1) and Q(x2,y2) plotted on the Cartesian plane.
Construct point R so that R has the same x-coordinate as Q and the same y-coordinate as P.
The coordinates of R are (x2,y1).
Triangle PQR is a right triangle with the right angle at R.
The horizontal side PR is the difference of x-coordinates in magnitude.
 PR = x2 - x1
The vertical side RQ is the difference of y-coordinates in magnitude.
RQ =y2 - y1

By the Pythagorean theorem applied to right triangle PQR:
PQ² = PR² + RQ².
Substitute the expressions for \(PR\) and \(RQ\):
PQ² = (x2 - x1)² + (y2 - y1)²

Distance is a non-negative quantity; therefore take the positive square root:Conclusion: The distance between two points P(x1,y1) and Q(x2,y2) in the plane is:

Remarks and special cases

• If the points have the same x-coordinate, i.e., x1 = x2, then the segment is vertical and

   PQ = y2 - y1.

• If the points have the same y-coordinate, i.e., y1 = y2, then the segment is horizontal and

  PQ = x2 - x1.

• Distance from the origin O(0,0) to a point P(x,y) is
• Symmetry: The distance is symmetric in its arguments; PQ = QP.
• Metric property: This formula defines the Euclidean distance (or geometric length) in the plane and satisfies the triangle inequality.

Worked example

Example: Find the distance between A(1,2) and B(4,6).

Compute the difference in x-coordinates.
x2 - x1 = 4 - 1 = 3.
Compute the difference in y-coordinates.
 y2 - y1 = 6 - 2 = 4.

Apply the distance formula:

Applications and uses

• Finding the length of a line segment between two coordinate points in geometry problems.
• Checking whether three points are vertices of a right triangle by comparing squared distances (Pythagorean test).
• Computing side lengths when using coordinate methods to find areas of polygons.
• Distance formula is used in analytic geometry, coordinate proofs, and in many real-life modelling problems requiring straight-line distances in a plane.

Understanding and applying the distance formula is a basic but powerful tool in coordinate geometry. Practice on several pairs of points, including integer and non-integer coordinates, to gain confidence in its use.