In this figure the square formed on the hypotenuse is divided into two rectangles. The additional side is determined by the division made in the hypotenuse by dropping a perpendicular line from the apex of the triangle to the hypotenuse. There are also two rhomboids in this frame, each of which has one side equal respectively to the large and to the small square of the sides of the triangle and the other side equal to the hypotenuse.
The shorter altitude of the two rhomboids, as may be seen in the figure itself, corresponds to the respective altitudes, or shorter sides, of the rectangles. But the longer side corresponds respectively to the side of the larger and of the smaller squares of the sides of the triangle.
It is not necessary that these corresponding dimensions be known by the child. He sees red and yellow pieces of an inset and simply moves them about, placing them in the indentures of the frame. It is the fact that these movable pieces actually fit into this white background which gives the child the opportunity for reasoning out the theorem, and not the abstract idea of the corresponding relations between the dimensions of the sides and the different heights of the figures. Reduced to these terms the exercise is easily performed and proves very interesting.
This material may be used for other demonstrations:
Demonstration A: The substitution of the pieces. Let us start with the frame as it should be filled originally. First take out the two rectangles formed on the hypotenuse; place them in the two lateral grooves, and lower the triangle. Fill the remaining empty space with the two rhomboids.
The same space is filled in both cases with:
A triangle plus two rectangles, and then
A triangle plus two rhomboids.
Hence the sum of the two rectangles (which form the square of the hypotenuse) is equal to the sum of the two rhomboids.
In a later substitution we consider the rhomboids instead of the rectangles in order to demonstrate their respective equivalence to the two squares formed on the sides of the triangle. Beginning, for example with the larger square, we start with the insets in the original position and consider the space occupied by the triangle and the larger square. To analyze this space the pieces are all taken out and then it is filled successively by:
The triangle and the large square in their original positions.
The triangle and the large rhomboid.