1. THE PYTHAGOREAN THEOREM AND THE PYTHAGOREAN ABACUS END Click or scroll down to begin

2. The Pythagorean Theorem states that given a right triangle with sides (a, b, and c), as shown, (a² + b²) will equal (c²). In other words the area of the yellow and red squares will fit into or equal the area of the orange square. A two dimensional proof of this follows that may be taught to fourth graders. First construct on each side of square (c) right triangles such that a larger square is formed with sides equal (a+b) Next construct right triangles on the inside of square (c) in the manner shown to the right: make the upper left red, the lower left yellow and the remaining two orange. This will demonstrate that the area of (c²) equals four times the area of the original right triangle plus the area of a square in its' middle. a b c a + b a b c 1 2 4 3 c c ? 1 4 3 2 Now outline a square with sides equal (b) in the upper right corner of the larger square and a square with sides equal (a) in the lower right corner of the larger square. This outlines the area (a²+b²). The Pythagorean Theorem states that this area should fit inside the (c) square. Well if we take a look at the orange area it is all inside of the (c) square and equals most of the area of (a²+b²) except for two blue right triangles, one in the upper right corner and one in the the lower right corner. It is easily seen that the area remaining inside the (c) square, the red and the yellow triangles, is exactly equal to the two blue right triangles. Therefore, the Pythagorean theorem is proved. a b

3. DEMONSTRATING THE PYTHAGOREAN THEOREM ON THE ABACUS Click screen or scroll down to begin

4. Let a=2 and b=3 then the area of the the large square equals (2+3)²=5². So multiply 5x5 on the abacus. It can be seen that the combined area of each two adjacent triangles constructed on the sides of (c²) equals the area of a rectangle with sides or factors equal (a and b). The checkered rectangle shows the area of these rectangles given a=2 and b=3). The square of beads on the abacus above can be arrayed to show these rectangles in a similar manner. The color coding shows which beads represent which area, three beads for each triangle, and the one bead remaining, in the middle, represents the area of the orange square. That the two rectangles on the right plus the middle square equals a²+b² is dramatically shown, when the middle bead, representing the middle square, is pushed to the right into the two rectangles.