Unformatted text preview: AB (that is, CD forms a right angle with AB ), then this divides △ ABC into two smaller triangles △ CBD and △ ACD , which are both similar to △ ABC . A C B b a c D d c − (a) △ ABC C D B d a (b) △ CBD A D C c − d b (c) △ ACD Figure 1.1.5 Similar triangles △ ABC , △ CBD , △ ACD Recall that triangles are similar if their corresponding angles are equal, and that similarity implies that corresponding sides are proportional. Thus, since △ ABC is similar to △ CBD , by proportionality of corresponding sides we see that AB is to CB (hypotenuses) as BC is to BD (vertical legs) ⇒ c a = a d ⇒ cd = a 2 . Since △ ABC is similar to △ ACD , comparing horizontal legs and hypotenuses gives b c − d = c b ⇒ b 2 = c 2 − cd = c 2 − a 2 ⇒ a 2 + b 2 = c 2 . QED Note: The symbols ⊥ and ∼ denote perpendicularity and similarity, respectively. For example, in the above proof we had CD ⊥ AB and △ ABC ∼△ CBD ∼△ ACD ....
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 Fall '11
 Dr.Cheun
 Calculus, Trigonometry, Pythagorean Theorem, Right triangle, Hypotenuse

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