Image source

The Pythagorean relationship is so important that it deserves some special attention. In geometric terms, this relationship states that if a square is constructed on each side of a right triangle, the areas of the two smaller squares will together equal the area of the square on the longest side, the hypotenuse. 

To discover this relationship, consider the following activity: 

Have students draw a right triangle on half-centimeter grid paper. Assign each student a different triangle by specifying the lengths of the two legs. Students are to draw a square on each leg and the hypotenuse and find the area of all three squares. (For the square on the hypotenuse, the exact area can be found by making each of the sides the diagonal of a rectangle.)

The image above shows Pythagorean relationship. Note that if drawn on a grid, the area of all squares is easily determined. Here 4 + 16 = area of the square on the hypotenuse. 

As an extension to the above activity, students can explore other figures on the legs of right triangles and computing areas. For example, draw semicircles or equilateral triangles instead of squares. The areas of any regular polygons drawn on the three sides of right triangles will have the same relationship.  

Image source

In the image above, tessellations on a grid helps in establishing the Pythagorean relationship. 

But what about proof?     

The image above shows two proofs that students can follow. The first consist of only the two drawings. An algebraic proof is shown below the drawings, based on the second square.