Final answer:
To prove the Pythagorean theorem, you can arrange four identical right triangles into a square within a square or two squares side by side. Both arrangements visually demonstrate that the sum of the squares of the legs (a² + b²) is equal to the square of the hypotenuse (c²).
Step-by-step explanation:
Proving the Pythagorean Theorem with Four Identical Triangles
To prove the Pythagorean theorem using four identical right triangles, we can arrange the triangles in two different shapes that will both provide a visual proof. One shape is a square within a square, and another is two squares side by side.
First Proof - Square within a Square
Imagine organizing the four triangles into a larger square such that they form the corners, with the hypotenuse of each triangle facing outwards. The outer square has a side length equal to a + b, where a and b are the legs of the triangles. The inner square has a side length equal to c, the hypotenuse. The area of the large square can be calculated in two ways: by the sum of the areas of the four triangles (4 * (1/2)ab) plus the area of the inner square (c2), or by the area of the large square ((a + b)2). Setting these equal to each other and simplifying proves a2 + b2 = c2.
Second Proof - Two Squares Side by Side
A second way to arrange the triangles is to place them around one square with side a, and then place the remaining triangles around another square with side b. The two squares together have a combined area of a2 + b2. The outer perimeter forms a square with side length c. Thus, the area of the large square formed by the hypotenuses is c2, providing the same result: a2 + b2 = c2.