35. Proof Coordinate Geometry You can use the Pythagorean Theorem to prove the Distance Formula. Let points P ( x 1 , y 1 p open , x sub 1 . comma , y sub 1 ) and Q ( x 2 , y 2 q open , x sub 2 . comma , y sub 2 ) be the endpoints of the hypotenuse of a right triangle.

    

    1. Write an algebraic expression to complete each of the following: P R = ? ¯ p r equals , modified question mark with macron below and Q R = ? ¯ . q r equals , modified question mark with macron below , .
    2. By the Pythagorean Theorem, P Q 2 = P R 2 + Q R 2 . p , q squared , equals , p , r squared , plus , q , r squared , . Rewrite this statement by substituting the algebraic expressions you found for PR and QR in part (a).
    3. Complete the proof by taking the square root of each side of the equation that you wrote in part (b).

Algebra Find the value of x. If your answer is not an integer, express it in simplest radical form.

For each pair of numbers, find a third whole number such that the three numbers form a Pythagorean triple.

39. 20, 21
40. 14, 48
41. 13, 85
42. 12, 37

Open-Ended Find integers j and k such that (a) the two given integers and j represent the side lengths of an acute triangle and (b) the two given integers and k represent the side lengths of an obtuse triangle.

43. 4, 5
44. 2, 4
45. 6, 9
46. 5, 10
47. 6, 7
48. 9, 12

49. Proof Prove the Pythagorean Theorem.

    

    Given: Δ A B C cap delta eh b c is a right triangle.

    Prove: a 2 + b 2 = c 2 eh squared , plus , b squared , equals , c squared

    (Hint: Begin with proportions suggested by Theorem 7-3 or its corollaries.)

50. Astronomy The Hubble Space Telescope orbits 600 km above Earth's surface. Earth's radius is about 6370 km. Use the Pythagorean Theorem to find the distance x from the telescope to Earth's horizon. Round your answer to the nearest ten kilometers. (Diagram is not to scale.)

    

51. Prove that if the slopes of two lines have product − 1 , negative 1 comma then the lines are perpendicular. Use parts (a)–(c) to write a coordinate proof.

    1. First, argue that neither line can be horizontal nor vertical.
    2. Then, tell why the lines must intersect. (Hint: Use indirect reasoning.)
    3. Place the lines in the coordinate plane. Choose a point on l 1 l sub 1 and find a related point on l 2 . l sub 2 , . Complete the proof.

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