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Pythagorean Theorem various visualizations Pythagorean Theorem • If this was part of a face-to-face lesson, I would cut out four right triangles for each pair of participants and ask you to discover these visualizations of why the Pythagorean Theorem is true. • Before you begin you might want to cut out four right triangles and play along! Pythagorean Theorem, I b a a c b c a+b c b c a+b a Area ( a b ) * ( a b ) 2 a ab ba b 2 2 Thus, 2 a b c 2 2 1 a*b 2 ab c a 2 ab b 2 ab c 2 + 2 must be equal 2 b + Area = a (a b) b(a b) a 2 ab b a 2 2 1 2 2 a*b 1 2 + a*b + 1 2 a *b c 2 Pythagorean Theorem, II a b a a a c b c b b b c c a a b Notice that each square has 4 dark green triangles. Therefore, the yellow regions must be equal. Yellow area a b 2 2 Yellow area c 2 Pythagorean Theorem, III aa b b-a b c b-a c a c c Area of whole square Area of whole square c*c c 2 a rea o f 4 g reen tria n g les a rea o f w h ite sq u a re 1 4 2 must be equal a b ( b a )( b a ) 2 ab b (b a ) a (b a ) 2 2 ab b ab ab a 2 2 ab b 2 ab a 2 b a 2 2 2 Pythagorean Theorem • The next demonstration of the Pythagorean Theorem involve cutting up the squares on the legs of a right triangle and rearranging them to fit into the square on the hypotenuse. This demonstration is considered a dissection. • I highly recommend paper and scissors for this proof of the Pythagorean Theorem. Pythagorean Theorem, IV • • Construct a right triangle. Construct squares on the sides. • Construct the center of the square on the longer leg. The center can be constructed by finding the intersection of the two diagonals. • Construct a line through the center of the square and parallel to the hypotenuse. Pythagorean Theorem, IV • Construct a line through the center of the square and perpendicular to the hypotenuse. • Now, you should have four regions in the square on the longer leg. The five interiors: four in the large square plus the one small square can be rearranged to fit in the square on the hypotenuse. This is where you will need your scissors to do this. • Once you have the five regions fitting inside the square on the hypotenuse, this should 2 2 2 illustrate that a b c