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How to Use the Pythagorean Theorem
The Pythagorean theorem is probably the most useful single formula in elementary math. It powers the distance formula, vector magnitudes, the trig unit circle, and 3D geometry. Memorising the common triples saves time on standardised tests.
When the method applies
- • Right triangle missing one side
- • Distance formula confusion
- • 3D problems with diagonal lengths
- • Coordinate geometry distance questions
Common mistakes
- • Mixing up legs vs hypotenuse
- • Squaring before adding (wrong)
- • Forgetting to take the square root
Step-by-step method
- • Formula: a² + b² = c² where c is the hypotenuse (longest side)
- • Solve for a leg: a = √(c² − b²)
- • Distance formula = Pythagorean theorem: d = √((x₂−x₁)² + (y₂−y₁)²)
- • 3D distance: d = √((Δx)² + (Δy)² + (Δz)²)
- • Memorise common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25
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Build long-term fluency
- • Always identify the hypotenuse first (opposite the right angle)
- • Sanity check: the hypotenuse is always the longest side
- • Draw the triangle even if the problem does not
Edge cases & deeper reading
For non-right triangles, use the Law of Cosines (the general version of Pythagoras). For curved surfaces or non-Euclidean geometries, the rule changes.
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