6 Chapter 1 • Right Triangle Trigonometry §1.1 10. One end of a rope is attached to the top of a pole 100 ft high. If the rope is 150 ft long, what is the maximum distance along the ground from the base of the pole to where the other end can be attached? You may assume that the pole is perpendicular to the ground. 11. Prove that the hypotenuse is the longest side in every right triangle. ( Hint: Is a 2 + b 2 > a 2 ? ) 12. Can a right triangle have sides with lengths 2, 5, and 6? Explain your answer. 13. The lengths of the sides of any right triangle form what is called a Pythagorean triple : positive integers a , b , and c such that a 2 + b 2 = c 2 . The triple is normally written as ( a , b , c ). For example, (3,4,5) and (5,12,13) are well-known Pythagorean triples. (a) Show that (6,8,10) is a Pythagorean triple. (b) Show that if ( a , b , c ) is a Pythagorean triple then so is ( ka , kb , kc ) for any integer k > 0. How would you interpret this geometrically? (c)
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