Area of Whole Square :

A = (a+b)(a+b)

Area of the smaller (tilted) square :

A= c²

Area of four triangles :

A=4(1/2 ab) = 2ab

The area of the large square is equal to the area of the tilted square and the four triangles.

(a+b)(a+b) = c² + 2ab

a² + 2ab +b² = c² + 2ab Expanding (a+b)(a+b)

a² + b² = c² Subtraction Property of Equality

Proof:

a² + b² = c²

The Pythagorean theoremcan be illustrated visually inseveral ways. One suchillustration is shown by thefigure on the right. Square Ais divided into four regionsby two dashed lines that passthrough its center. Onedashed line is parallel to theleft edge of square C, and theother dashed line is parallelto the lower edge of squareC. If these four regions andsquare B are traced and cutout, they can be arranged tocover square C. Try it!

Starting Points for Investigations

1. Draw a right triangle with legs of length 1 inch and 2inches and a square on each of its sides. (Right anglescan be drawn by using the corner of a file card.)Subdivide the larger of the squares on the legs into fourregions as described in the figure on the previous slide.Show how these four regions and the square on the otherleg can be arranged to exactly cover the square on thehypotenuse.

2. Suppose the smaller of the squares on the legs of aright triangle is subdivided into four regions as describedabove. Can these regions and the square on the other legbe arranged to cover the square on the hypotenuse?