The Pythagorean Theorem: a² + b² = c²
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. Written as a formula: a² + b² = c², where c is the hypotenuse and a and b are the two legs.
This theorem is named after the ancient Greek mathematician Pythagoras (c. 570-495 BCE), although evidence suggests that Babylonian mathematicians knew the relationship over a thousand years earlier. It is one of the most fundamental results in all of mathematics and has been proven in hundreds of different ways — more than any other theorem.
The theorem only works for RIGHT triangles — triangles that have exactly one 90-degree angle. The hypotenuse is always the longest side and is always opposite the right angle. If you are working with a non-right triangle, you need the law of cosines instead: c² = a² + b² - 2ab cos(C).
How to Use the Pythagorean Theorem
The most common use is finding a missing side of a right triangle when you know the other two sides. There are two scenarios: finding the hypotenuse, or finding a leg.
Finding the hypotenuse: If a = 3 and b = 4, then c² = 3² + 4² = 9 + 16 = 25, so c = √25 = 5. This is the famous 3-4-5 right triangle.
Finding a leg: If c = 13 and a = 5, then b² = c² - a² = 169 - 25 = 144, so b = √144 = 12. This is the 5-12-13 right triangle.
When the answer is not a perfect square, leave it in simplified radical form or use a calculator to get a decimal approximation. For example, if a = 2 and b = 3, then c = √(4 + 9) = √13 ≈ 3.606.
Pythagorean Triples
A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a² + b² = c². Knowing common triples saves time on tests because you can recognize them instantly without calculation.
The most common Pythagorean triples are: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41. Any multiple of a triple is also a triple: 6-8-10 (double of 3-4-5), 10-24-26 (double of 5-12-13), and so on.
You can generate Pythagorean triples using the formula: a = m² - n², b = 2mn, c = m² + n², where m and n are positive integers with m > n. For example, m = 2, n = 1 gives (3, 4, 5). m = 3, n = 2 gives (5, 12, 13). This formula generates all primitive triples when m and n are coprime and not both odd.
A Visual Proof of the Pythagorean Theorem
One of the most elegant proofs uses area. Imagine a large square with side length (a + b). Inside it, arrange four copies of the right triangle to form a smaller tilted square in the center whose side length is c.
The area of the large square is (a + b)² = a² + 2ab + b². The area can also be computed as the four triangles plus the inner square: 4 × (½ab) + c² = 2ab + c². Setting these equal: a² + 2ab + b² = 2ab + c². Subtract 2ab from both sides: a² + b² = c². This proof was known to the ancient Chinese and appears in the Zhoubi Suanjing, a mathematical text from around 100 BCE.
The Distance Formula: Pythagorean Theorem in Disguise
The distance formula used in coordinate geometry is a direct application of the Pythagorean theorem. To find the distance between two points (x₁, y₁) and (x₂, y₂), draw a right triangle where the horizontal leg is |x₂ - x₁| and the vertical leg is |y₂ - y₁|. The distance is the hypotenuse: d = √((x₂ - x₁)² + (y₂ - y₁)²).
This extends to three dimensions: d = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²). The same principle underlies how GPS systems calculate distances and how video games compute positions of objects in 3D space.
Real-World Applications
Construction: Builders use the 3-4-5 triangle to verify right angles. Measure 3 feet along one wall, 4 feet along the other, and if the diagonal measures exactly 5 feet, the corner is perfectly square.
Navigation: Pilots and sailors calculate the shortest distance between two points using the Pythagorean theorem on coordinate grids. The straight-line distance (as the crow flies) is the hypotenuse of a triangle formed by east-west and north-south distances.
Screen sizes: When manufacturers say a TV is 55 inches, they mean the diagonal of the screen. If the screen is 48 inches wide and 27 inches tall, you can verify: √(48² + 27²) = √(2304 + 729) = √3033 ≈ 55.07 inches.
Ladder safety: If you lean a 10-foot ladder against a wall and the base is 6 feet from the wall, the ladder reaches √(100 - 36) = √64 = 8 feet up the wall. Safety guidelines recommend the base be about one-quarter of the working height, so this setup (6 feet out, 8 feet up) is within guidelines.
The Converse of the Pythagorean Theorem
The converse states: if a² + b² = c² for three sides of a triangle, then the triangle is a right triangle. This gives you a way to TEST whether a triangle is a right triangle without measuring angles.
You can also determine the type of triangle: if a² + b² > c², the triangle is acute (all angles less than 90 degrees). If a² + b² < c², the triangle is obtuse (one angle greater than 90 degrees). Always make sure c is the longest side before making this comparison.
Example: is a triangle with sides 6, 8, and 11 a right triangle? Check: 6² + 8² = 36 + 64 = 100. 11² = 121. Since 100 < 121, this is an obtuse triangle, not a right triangle.
Common Mistakes and Tips
The biggest mistake is confusing which side is the hypotenuse. Remember: the hypotenuse is ALWAYS opposite the right angle and ALWAYS the longest side. If your calculated side turns out longer than what you labeled as the hypotenuse, you have made an error.
Another common error is using the theorem on non-right triangles. Always verify that the triangle has a 90-degree angle before applying a² + b² = c². For non-right triangles, use the law of cosines.
When you get an answer like √50, simplify it: √50 = √(25 × 2) = 5√2. Many teachers require simplified radicals, and it is a common place to lose points on exams.
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