Pre-Calc Exam Notes 4

Pre-Calc Exam Notes 4 - AB (that is, CD forms a right angle...

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4 Chapter 1 Right Triangle Trigonometry §1.1 A C B b a c Figure 1.1.4 In a right triangle, the side opposite the right angle is called the hy- potenuse , and the other two sides are called its legs . For example, in Figure 1.1.4 the right angle is C , the hypotenuse is the line segment AB , which has length c , and BC and AC are the legs, with lengths a and b , respectively. The hypotenuse is always the longest side of a right triangle (see Exercise 11). By knowing the lengths of two sides of a right triangle, the length of the third side can be determined by using the Pythagorean Theorem : Theorem 1.1. Pythagorean Theorem: The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of its legs. Thus, if a right triangle has a hypotenuse of length c and legs of lengths a and b , as in Figure 1.1.4, then the Pythagorean Theorem says: a 2 + b 2 = c 2 (1.1) Let us prove this. In the right triangle ABC in Figure 1.1.5(a) below, if we draw a line segment from the vertex C to the point D on the hypotenuse such that CD is perpendicular to
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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 exam-ple, in the above proof we had CD AB and ABC CBD ACD ....
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This note was uploaded on 01/21/2012 for the course MAC 1130 taught by Professor Dr.cheun during the Fall '11 term at FSU.

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