The Pythagorean Theorem

Statement of the Theorem

The Pythagorean theorem is one of the oldest and most widely used results in all of mathematics. It describes a fundamental relationship between the three sides of any right triangle: the square of the hypotenuse is equal to the sum of the squares of the other two sides.

If a right triangle has legs of length a and b, and a hypotenuse of length c, then:

The hypotenuse is always the longest side of a right triangle and is the side directly opposite the right angle. The other two sides are called legs. This theorem only applies to right triangles — triangles that contain exactly one 90-degree angle.

Visual Proof: The Rearrangement Method

There are over 370 known proofs of the Pythagorean theorem, but the rearrangement proof is one of the most intuitive and elegant. Here is how it works:

Imagine you have a large square with side length (a + b). Inside this square, arrange four identical copies of your right triangle (each with legs a and b and hypotenuse c) so that their hypotenuses form a smaller, tilted square in the center.

1. Total area of the big square: (a + b)² = a² + 2ab + b²
2. Area from the four triangles: 4 × (1/2 × a × b) = 2ab
3. Area of the inner square: c²
4. Setting them equal: a² + 2ab + b² = 2ab + c²
5. Subtract 2ab from both sides: a² + b² = c²

This proof requires no algebra beyond basic area formulas. The four triangles account for 2ab of the total area, and whatever is left must be c². Since the big square has area a² + 2ab + b², subtracting the 2ab from the triangles leaves a² + b² = c².

Algebraic Proof Using Similar Triangles

Drop a perpendicular from the right angle to the hypotenuse, creating two smaller triangles inside the original. Each of these smaller triangles is similar to the original triangle and to each other.

Let the altitude have length h, and let it divide the hypotenuse c into segments of length p and q, where p + q = c. By the similarity of the triangles:

• From the first pair of similar triangles: a² = p × c
• From the second pair: b² = q × c

Adding these two equations: a² + b² = pc + qc = (p + q)c = c × c = c². This proof is particularly beautiful because it uses only the properties of similar triangles and requires no area arguments.

Pythagorean Triples

A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a² + b² = c². These are special because all three sides are whole numbers, making calculations clean and exact.

Any multiple of a Pythagorean triple is also a Pythagorean triple. For example, if (3, 4, 5) is a triple, then so is (6, 8, 10), (9, 12, 15), (30, 40, 50), and so on. The triple (3, 4, 5) is by far the most commonly encountered and is worth memorizing.

The Converse of the Pythagorean Theorem

The converse of the Pythagorean theorem states: if the square of the longest side of a triangle equals the sum of the squares of the other two sides, then the triangle is a right triangle. This is extremely useful for testing whether a triangle contains a 90-degree angle.

More broadly, you can classify any triangle by comparing c² with a² + b²:

• If c² = a² + b², the triangle is a right triangle
• If c² < a² + b², the triangle is an acute triangle (all angles less than 90°)
• If c² > a² + b², the triangle is an obtuse triangle (one angle greater than 90°)

Connection to the Distance Formula

The distance formula from coordinate geometry is a direct application of the Pythagorean theorem. To find the distance between two points (x₁, y₁) and (x₂, y₂), you form a right triangle where the horizontal distance is |x₂ − x₁| and the vertical distance is |y₂ − y₁|. The distance is the hypotenuse:

This is simply a² + b² = c² solved for c, with a = |x₂ − x₁| and b = |y₂ − y₁|. Every time you compute a distance in a coordinate plane, you are applying the Pythagorean theorem.

Extension to Three Dimensions

The Pythagorean theorem extends naturally to three-dimensional space. If you need the distance between points (x₁, y₁, z₁) and (x₂, y₂, z₂), the formula becomes:

This works because you can apply the Pythagorean theorem twice: first in the xy-plane to get the horizontal distance, then again combining that horizontal distance with the vertical z-component.

Real-World Applications

The Pythagorean theorem appears in an enormous variety of practical situations:

• Construction and architecture: Builders use the 3-4-5 rule to verify that corners are square. By measuring 3 feet along one wall and 4 feet along the adjacent wall, the diagonal should be exactly 5 feet if the corner is a true 90-degree angle.
• Navigation: Pilots, sailors, and hikers calculate straight-line distances between waypoints using the theorem, treating north-south and east-west distances as legs of a right triangle.
• Screen sizes: Television and monitor sizes are measured diagonally. If a screen is 16 inches wide and 9 inches tall, the diagonal is √(16² + 9²) = √(256 + 81) = √337 ≈ 18.36 inches.
• Surveying and mapping: Land surveyors calculate distances across terrain features by breaking them into right-triangle components.
• Physics: Vector magnitude calculations rely on the Pythagorean theorem. If a force has components of 3N east and 4N north, its magnitude is 5N.

Worked Examples

Practice Problems