SECTION 1.2 The Distance and Midpoint Formulas 17 ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 1.2 Assess Your Understanding 1. If −3 and 5 are the coordinates of two points on the real number line, the distance between these points is . (pp. A5–A6) 2. If 3 and 4 are the legs of a right triangle, the hypotenuse is . (p. A14) 3. Use the converse of the Pythagorean Theorem to show that a triangle whose sides are of lengths 11, 60, and 61 is a right triangle. (pp. A14–A15) 4. In the rectangular coordinate system, the origin has coordinates . (p. 2) 5. The area A of a triangle whose base is b and whose altitude is h is = A . (p. A15) 6. True or False Two triangles are congruent if two angles and the included side of one equals two angles and the included side of the other. (pp. A16–A17) Concepts and Vocabulary 7. If three distinct points P, Q, and R all lie on a line, and if ( ) ( ) = d P Q d Q R , , , then Q is called the of the line segment from P to R. 8. True or False The distance between two points is sometimes a negative number. 9. True or False The midpoint of a line segment is found by averaging the x -coordinates and averaging the y -coordinates of the endpoints. 10. Multiple Choice Choose the expression that equals the distance between two points ( ) x y , 1 1 and ( ) x y , . 2 2 (a) ( ) ( ) − + − x x y y 2 1 2 2 1 2 (b) ( ) ( ) + − + x x y y 2 1 2 2 1 2 (c) ( ) ( ) − − − x x y y 2 1 2 2 1 2 (d) ( ) ( ) + + + x x y y 2 1 2 2 1 2 Skill Building In Problems 11–24, find the distance d between the points P1 and P .2 11. P1 = (0, 0) P2 = (2, 1) x y –2 –1 2 2 12. P1 = (0, 0) P2 = (–2, 1) x y –2 –1 2 2 13. –1 x y 2 2 –2 P2 5 (–2, 2) P1 5 (1, 1) 14. x y –2 –1 2 2 P 1 = (–1, 1) P2 = (2, 2) 15. ( ) ( ) = − = P P 3, 4; 5, 4 1 2 16. ( ) ( ) = − = P P 1, 0 ; 2, 4 1 2 17. ( ) ( ) = − = P P 7, 3 ; 4, 0 1 2 18. ( ) ( ) = − = P P 2, 3; 4, 2 1 2 19. ( ) ( ) = − = P P 5, 2; 6, 1 1 2 20. ( ) ( ) = − − = P P 4, 3; 6, 2 1 2 21. ( ) ( ) = − = P P 0.2, 0.3 ; 2.3, 1.1 1 2 22. ( ) ( ) = = − P P 1.2, 2.3 ; 0.3, 1.1 1 2 23. ( ) ( ) = = P a b P , ; 0, 0 1 2 24. ( ) ( ) = = P a a P , ; 0, 0 1 2 In Problems 25–30, plot each point and form the triangle ABC. Show that the triangle is a right triangle. Find its area. 25. ( ) ( ) ( ) = − = = − A B C 2, 5 ; 1, 3 ; 1, 0 26. ( ) ( ) ( ) = − = = − A B C 2, 5 ; 12, 3 ; 10, 11 27. ( ) ( ) ( ) = − = = A B C 5, 3 ; 6, 0 ; 5, 5 28. ( ) ( ) ( ) = − = − = − A B C 6, 3 ; 3, 5; 1, 5 29. ( ) ( ) ( ) = − = − = A B C 4, 3; 0, 3; 4, 2 30. ( ) ( ) ( ) = − = = A B C 4, 3; 4, 1 ; 2, 1 In Problems 31–38, find the midpoint of the line segment joining the points P1 and P .2 31. ( ) ( ) = − = P P 3, 4; 5, 4 1 2 32. P P 2, 0 ; 2, 4 1 2 ( ) ( ) = − = 33. ( ) ( ) = − = P P 1, 4 ; 8, 0 1 2 34. ( ) ( ) = − = P P 2, 3; 4, 2 1 2 35. ( ) ( ) = − = P P 7, 5; 9, 1 1 2 36. ( ) ( ) = − − = P P 4, 3; 2, 2 1 2 37. ( ) ( ) = = P a b P , ; 0, 0 1 2 38. ( ) ( ) = = P a a P , ; 0, 0 1 2 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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