The Pythagorean theorem is one of the most famous theorems in mathematics. It states that in a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. Here we see a right triangle with legs labeled 'a' and 'b', and hypotenuse labeled 'c'.
The mathematical formula for the Pythagorean theorem is a squared plus b squared equals c squared. Here, 'a' and 'b' represent the lengths of the two legs of the right triangle, while 'c' represents the length of the hypotenuse. The squares shown below illustrate this relationship visually - the area of the square on the hypotenuse equals the sum of the areas of the squares on the other two sides.
Let's work through a classic example. We have a right triangle where one leg is 3 units and the other leg is 4 units. To find the hypotenuse, we use the Pythagorean theorem. We substitute our values: c squared equals 3 squared plus 4 squared, which gives us c squared equals 9 plus 16, equals 25. Taking the square root, we get c equals 5. This is the famous 3-4-5 triangle, one of the most well-known Pythagorean triples.
Here's a beautiful visual proof of the Pythagorean theorem. We construct squares on each side of our right triangle. The square on side 'a' has area a squared, the square on side 'b' has area b squared, and the square on the hypotenuse has area c squared. Through geometric rearrangement, we can show that the area of the large square on the hypotenuse exactly equals the combined areas of the two smaller squares. This visual demonstration makes the relationship a squared plus b squared equals c squared immediately clear.
勾股定律，也被称为毕达哥拉斯定理，是几何学中最著名的定理之一。它描述了直角三角形中三边之间的关系：在直角三角形中，两个直角边的平方和等于斜边的平方。这个定理可以用图形直观地理解——以三角形三边为边长的正方形，其中两个较小正方形的面积之和等于最大正方形的面积。
勾股定律的数学表达式为：a² + b² = c²。这里a和b代表直角三角形的两条直角边，c代表斜边，也就是直角对面的最长边。让我们用一个具体的例子来验证：当a=3，b=4时，根据勾股定律，c应该等于5。计算验证：3²+4²=9+16=25，而5²=25，所以等式成立。
现在让我们看一下勾股定律的几何证明。这个证明使用面积方法：我们构造一个边长为a+b的大正方形，它可以分解成四个相同的直角三角形和一个边长为c的内正方形。大正方形的面积等于(a+b)²，展开得到a²+2ab+b²。同时，这个面积也等于四个三角形的面积加上内正方形的面积，即4×(½ab)+c²=2ab+c²。因此我们得到：a²+2ab+b²=2ab+c²，消去2ab后得到a²+b²=c²。
勾股数是指满足勾股定律的三个正整数组合。最著名的勾股数是(3,4,5)，但还有许多其他的勾股数，比如(5,12,13)、(8,15,17)、(7,24,25)和(20,21,29)等。让我们验证(5,12,13)：5²+12²=25+144=169，而13²=169，所以等式成立。这些整数组合在古代建筑和测量中非常实用，因为可以用简单的整数长度构造精确的直角。
勾股定律有着广泛的实际应用。在建筑和工程中，它帮助计算距离和确保结构的准确性。导航系统和GPS使用它来确定最短路径。计算机图形学依靠它进行距离计算和三维渲染。工程师在设计和分析中使用它。从古代到现代科技，这个简单而强大的直角三角形三边关系仍然是数学中最有用和最基本的定理之一。