The Pythagorean theorem is one of the most famous principles in mathematics. It describes the relationship between the sides of a right triangle. The formula states that a squared plus b squared equals c squared, where a and b are the legs of the triangle, and c is the hypotenuse - the longest side opposite the right angle.
We can visualize the Pythagorean theorem by drawing squares on each side of the right triangle. The red square has area a squared, the green square has area b squared, and the yellow square has area c squared. The theorem states that the sum of the areas of the two smaller squares equals the area of the largest square.
Let's solve a practical example. A ladder is 5 meters long and leans against a wall. The bottom of the ladder is 3 meters from the wall. How high up the wall does the ladder reach? Using the Pythagorean theorem: 3 squared plus b squared equals 5 squared. This gives us 9 plus b squared equals 25, so b equals 4 meters.
Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem. The most famous example is 3, 4, 5, where 3 squared plus 4 squared equals 5 squared, or 9 plus 16 equals 25. Other common triples include 5, 12, 13 and 8, 15, 17. These whole number solutions have been known for thousands of years.
To summarize what we have learned: The Pythagorean theorem is a fundamental principle that applies only to right triangles. The formula a squared plus b squared equals c squared allows us to find unknown side lengths. This theorem has practical applications in construction, navigation, and engineering, and forms the foundation for more advanced mathematical concepts.