The Pythagorean Theorem - Educational Outreach

Define a new function21 A2 A3 2xcp 2cxcp 2g( x , y) Aycpfor all points y on the vertical line segment where the stickfigure is walking. The function g( x cp, y)is clearlycontinuous on the closed line segment y : y [0, c] wherey cp ( 0, c). We note the following three results:1. g( x ,0) 2x( x c) 0cpcpcp2cp2. g(x , y ) 2x[ c x ] y 0cpcp2cp3. g ( x , c) 2x 2c(c x ) 0 .cpcpBy continuity, we can immediately deduce that our functiong( x cp, y)has the following behavior directly associated withthe three results as given above:1. ( x , y) 0g cpfor all y 0, y )cpcp[cpThis in turn implies that A1 A2 A3.2. g( , ) 0x cpy cpThis in turn implies that A1 A2 A3.3. ( x , y) 0g cpfor all y ( y , c]This in turn implies that A1 A2 A3.Tying these results to the obvious measure of the associatedinterior vertex angle in Figure 2.37 leads to our finalrevised statement of the Pythagorean Theorem and converseon the next page.cp281