| Pythagorean Theorem | |
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| The figure above represents a farm with four tenant farmers.
Each tenant farmer has a rectangle plot of land composed of a triangle of farm
land and a triangle of grazing land. In the middle of these farm plots is a shared
square of grazing land with a watering pond in its center. Lets designate the
shorter sides of the rectangles (A), the longer sides (B) and the diagonals dividing
the rectangle into two triangles (C). Then, the area including the pond and grazing land
bordered by the diagonals will equal C-squared. The tenant farmer of the upper
right hand rectangle is the farm owners land master. He monitors the growth of the crops of the other tenants and collects from them
part of what they grow in trade for the use of the land and watering pond. Pressing the
enter key will allow you to use the arrow keys to take the land master to each plot of land.
After leaving the land masters plot, walk counter clockwise and visit each of the plots in
turn. The corner stones will turn to point your way. Notice on the simulation of the Pythabacus the 4 times 4 quadrilateral; it represents the square of farm land illustrated above. As the land master walks from farm plot to farm plot, columns of beads will separate from the quadrilateral to represent each plot. The (A) length of each plot will be one bead and the (B) length of each plot will be three beads. The Pythagorean Theorem states that lengths A-squared plus B-squared will equal C-squared. |
When the land master reached the last tenant, he found that the last tenant
did not grow enough crops to trade for the use of the land and the watering
pond. The land master decided to take part of the last tenants land to grow crops for himself. He left the last tenant only a small plot equal A-squared parts of the whole farm. If you include the square of watering land
the land master now controls B-squared parts of the whole farm. Each rectangle equals two triangles and two rectangles equal four triangles
If you look closely at the C-squared part of the farm you will see it is bordered by four triangles with the watering square in the middle. So C-squared is
equal two rectangles plus the watering square. If you now look closely at the
A-squared part plus the B-squared part of the whole farm, you will see that these
parts also equal two rectangles and the watering square. Therefore A-squared
plus B-squared equal C-squared. On the Pythabacus simulation the four beads in the middle equal the watering
square and the columns of beads to its right equal two rectangles. Now walk
the land master through the gate and into the A-squared part of the farm. The watering square beads plus the two rectangle columns became two groups of beads equal A-squared and B-squared. You have proven the Pythagorean Theorem on the Pythabacus! So, the cows now drink happily at the pond. |