Derivation/Proof of Theorems: Coordinate Geometry | Mathematics (Maths) Class 10 PDF Download

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Theorem: Distance Formula 

Statement: In a two-dimensional Cartesian plane, the distance between two points P(x₁, y₁) and Q(x₂, y₂) is given by the formula:

PQ = √[(x₂ - x₁)² + (y₂ - y₁)²]

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

To Prove: The distance PQ between the points P and Q is equal to √[(x₂ - x₁)² + (y₂ - y₁)²].

 Proof:

• Plot points P(x₁, y₁) and Q(x₂, y₂) on the Cartesian plane.

• Draw a horizontal line from P to a point R directly below or above Q, such that R shares the same x-coordinate as Q and the same y-coordinate as P. Thus, R has coordinates (x₂, y₁).

• This forms a right-angled triangle PQR with:

  • Horizontal side PR of length |x₂ - x₁|
  • Vertical side RQ of length |y₂ - y₁|

• According to the Pythagorean theorem:

  PQ² = PR² + RQ²

• Substituting the lengths of PR and RQ:

  PQ² = (x₂ - x₁)² + (y₂ - y₁)²

• Taking the positive square root of both sides to represent distance:

  PQ = √[(x₂ - x₁)² + (y₂ - y₁)²]

Conclusion:

The distance between two points P(x₁, y₁) and Q(x₂, y₂) in a Cartesian plane is √[(x₂ - x₁)² + (y₂ - y₁)²], as derived using the Pythagorean theorem.

This formula is fundamental in coordinate geometry for calculating the straight-line distance between two points.