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MathidiaMathematics Academy
Coordinate geometry
Lesson 39 of 75
Challenge

Coordinate proof

Objective. Use algebraic distances and gradients to prove geometric properties.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Coordinate proof uses algebraic expressions for gradients and distances to establish geometric properties, such as showing a shape is a parallelogram or a triangle is right-angled. To prove two lines are perpendicular, show their gradients multiply to −1; to prove sides are equal, use the distance formula √((x2−x1)²+(y2−y1)²). For a triangle with vertices A(0,0), B(4,0), C(4,3), gradient of AB is 0 and gradient of BC is undefined (vertical), confirming a right angle at B. Always state explicitly which property (equal gradients, perpendicular gradients, or equal lengths) proves the claim. Writing exact values rather than decimals keeps proofs rigorous and avoids rounding errors.

Worked example

Prove that the line segment from A(1, 2) to B(5, 2) is perpendicular to the segment from B(5, 2) to C(5, 8).

1 … 2
Illustrated card showing 1 … 2
  1. 1Gradient of AB = (2−2)/(5−1) = 0/4 = 0
  2. 2Gradient of BC = (8−2)/(5−5) = 6/0, which is undefined (vertical line)
  3. 3A horizontal line (gradient 0) and a vertical line (undefined gradient) always meet at a right angle
  4. 4Therefore AB is perpendicular to BC

Answer: AB is perpendicular to BC (horizontal meets vertical)

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Common mistake. Pupils estimate lengths from a sketch instead of calculating with the distance formula, leading to unreliable proofs. For points (1,1) and (4,5), they might guess the length is "about 5" rather than computing √(3²+4²)=5 exactly. Always use exact coordinate formulas, not visual estimates.

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