1a+-+Bhaskara+II+(1114+-+1185)

Legend has it that this figure (well, a similar black and white version) is shown in a manuscript by Hindu mathematician Bhaskara II (1114 - 1185) accompanied only by the word 'Behold!' The proof was considered so obvious, no more needed to be said. Whether 'Behold!' actually accompanied the proof or not, it is a fairly easy proof to understand and one that is still often used in middle school and high school. Many textbooks refer to it as the "tilted square" proof.

media type="custom" key="6286003" Created using GeoGebra 3.2.40.0 GeoGebra - Dynamic Mathematics for Everyone []

1. Look at the figure. Bhaskara II reportedly found it so obvious, his only comment on the visual proof was "Behold!" Can you prove the Pythagorean Theorem from the picture?

2. Click on show variables so our assignment of a, b, and c is consistent. Notice there are four right triangles. Consider how the areas of the squares and triangles is related. Does this suggest a solution?

3. What is the formula for the area of the upright square in terms of a and b?

4. What is the area of the four right triangles in terms of a and b?

5. The area of the tilted square is c 2. Now that you have determined the areas of the upright square and triangles, express the area of the tilted square in terms of a and b. Does this equation look familiar?

6. Click on show areas. Move the slider bars for the side length upright square until the length is seven units. Now move the slider bar for the side length of the tilted square until the vertex of the green square divides the edge of the blue square into two segments, one of length 3 and one of length 4. Calculate the area of the tilted square using the method just defined: Area of Tilted Square = Area of Upright Square - Area of 4 Triangles. Does this confirm the theorem? Why or why not?