This proof was discovered by President J.A. Garfield in 1876 . The key is the formula for the area of a trapezoid ? half sum of the bases times the altitude ? ½ * (a+b) * (a+b). Looking at the picture another way, this also can be computed as the sum of areas of the three triangles ? ½*a*b + ½*a*b + ½*c*c. As before, simplifications yield a2+ b2=c2. Here is the following calculation. ½(a + b)(a + b) = ½ab + ½ab + ½cc ½(a + b)2 = ½(ab + ab + cc) (a + b)2 = (ab + ab + cc) a2 + b2 + 2ab = 2ab + c2 a2 + b2 = c2 EXAMPLES: Find the unknown variable 4 cm 7cm d Solution: d2 + 42=72 d2 = 49 - 16 d = 5.74 cm d x 13cm 5cm Solution: d2 = 132 - 52 d2 = 169 - 25 d2 = 144 d = 12 cm Solve for x x2 = 122 +122 x2=144+144 x2 = 288 x = 17.0 cm Problem Analysis: Find the length of a diagonal of a rectangle of length 9 cm and width 4 cm. 4 cm 9 cm Solution: d2 = 92 + 42 d2 = 81 + 16 d2 = 97 d = 9.85 cm A square has diagonals of length 10 cm. Find the sides of the square. 10 cm s2 + s2 = 102 2s2 = 100 s2 = 50 s = 7.07 cm A ship sails 20 km due North and then 35 km due East. How far is it from its starting point? 20km 35 km x Solution: X2 = 202 + 352 X2 = 400 + 1225 X2 = 1625 X = 40.3 km DRILL: A 4 m ladder rests against a vertical wall with its foot 2 m from the wall. How far up the wall does the ladder reach? 2. Find the length of a diagonal of a rectangular box of length 12 cm, width 5 cm and height 4 cm. ?It is better wither to be silent, or to say things of more value than silence. Sooner throw a pearl at hazard than an idle or useless word; and do not say a little in many words, but a great deal in a few. ? -Pythagoras
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