Understanding the Pythagorean Theorem

The Pythagorean theorem is one of the oldest and most useful results in mathematics. It applies only to right-angled triangles and relates the three sides.

The theorem

In a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c². The hypotenuse is always the longest side.

Finding the hypotenuse

If a = 3 and b = 4, then c² = 9 + 16 = 25, so c = 5. This is the famous 3-4-5 triangle, the simplest Pythagorean triple.

Finding a leg

If you know the hypotenuse and one leg, rearrange: b = √(c² − a²). If c = 13 and a = 5, then b = √(169 − 25) = √144 = 12.

Common Pythagorean triples

Memorize these: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (9, 40, 41). Multiples of these also work: (6, 8, 10) is just 2×(3, 4, 5). Recognizing triples saves time on exams.

Real-world applications

The theorem is used in construction (checking square corners), navigation (shortest distance), computer graphics (distance between pixels), and physics (vector magnitudes). Any time you need a straight-line distance, the Pythagorean theorem applies.

The distance formula connection

The distance formula between two points √((x₂−x₁)² + (y₂−y₁)²) is just the Pythagorean theorem applied to the horizontal and vertical distances. The distance is the hypotenuse.

Practice with the Math Champion geometry formulas, or try the algebra quiz for equation-solving practice.