Welcome to our exploration of the Pythagorean Theorem! This fundamental theorem in geometry reveals a beautiful relationship between the sides of a right triangle. Here we see a right triangle with sides a, b, and c, where c is the hypotenuse. The colored squares show the areas of each side squared.
The Pythagorean Theorem can be expressed with the simple formula: a squared plus b squared equals c squared. Let's verify this with our example triangle where a equals 3, b equals 4, and c equals 5. Substituting these values: 3 squared plus 4 squared equals 5 squared, which gives us 9 plus 16 equals 25. Indeed, 25 equals 25, confirming our theorem!
Now let's apply the Pythagorean Theorem to solve a real-world problem. We have a ladder that is 10 feet long leaning against a wall, with its base 6 feet from the wall. We need to find how high the ladder reaches. Using our formula: a squared plus b squared equals c squared. We know c equals 10 and a equals 6, so we need to find b. Substituting: 6 squared plus b squared equals 10 squared, which gives us 36 plus b squared equals 100. Solving for b squared: b squared equals 64, so b equals 8 feet.
Pythagorean triples are special sets of three positive integers that perfectly satisfy the Pythagorean theorem. The most well-known triple is 3, 4, 5, where 3 squared plus 4 squared equals 5 squared: 9 plus 16 equals 25. Other common triples include 5, 12, 13 and 8, 15, 17. These triples are incredibly useful in construction, navigation, and many practical applications where right angles are needed.
The Pythagorean Theorem has countless real-world applications that we encounter every day. In construction and architecture, builders use it to ensure structures are square and to calculate roof dimensions. GPS navigation systems use it to determine distances and positions. Computer graphics rely on it for rendering and distance calculations. Engineers use it in design work, and it's even used in sports for field layouts and equipment design. This ancient theorem, discovered over 2500 years ago, continues to be one of the most practical and useful tools in mathematics and science today.