Applications of Coordinate Geometry

This lesson brings together the techniques from the entire coordinate geometry topic and applies them to extended problems. The Edexcel 9MA0 specification tests these skills in multi-step questions that combine several areas of knowledge.

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Application 1: Properties of Geometric Shapes

Coordinate geometry can prove properties of triangles, quadrilaterals and other shapes.

Proving a Triangle Is Right-Angled

Show that two sides are perpendicular (product of gradients = −1).

Example: Show that the triangle with vertices A(1, 2), B(4, 6) and C(8, 3) is right-angled.

Gradient AB = (6 − 2)/(4 − 1) = 4/3. Gradient BC = (3 − 6)/(8 − 4) = −3/4.

AB × BC = (4/3)(−3/4) = −1. So AB ⊥ BC, and the triangle is right-angled at B.

Proving a Quadrilateral Is a Parallelogram

Show that opposite sides are parallel (equal gradients).

Example: Show that A(0, 0), B(4, 1), C(5, 4), D(1, 3) form a parallelogram.

Gradient AB = 1/4. Gradient DC = (4 − 3)/(5 − 1) = 1/4. AB ∥ DC ✓. Gradient AD = 3/1 = 3. Gradient BC = (4 − 1)/(5 − 4) = 3. AD ∥ BC ✓.

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