precalc midpoint formulas - 6 CHAPTER 1 Graphs THEOREM Midpoint Formula The midpoint M = 1x y2 of the line segment from P1 = 1x1 y12 to P2 = 1x2 y22 is

precalc midpoint formulas - 6 CHAPTER 1 Graphs THEOREM...

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6 CHAPTER 1 Graphs THEOREM Midpoint Formula Themidpoint ofthelinesegmentfrom to is (2) M = 1 x , y 2 = ¢ x 1 + x 2 2 , y 1 + y 2 2 P 2 = 1 x 2 , y 2 2 P 1 = 1 x 1 , y 1 2 M = 1 x , y 2 Finding the Midpoint of a Line Segment Find the midpoint of the line segment from to . Plot the points and and their midpoint. P 2 P 1 P 2 = 1 3, 1 2 P 1 = 1 - 5, 5 2 EXAMPLE 4 Solution Apply the midpoint formula (2) using and . Then the coordinates of the midpoint M are That is, . See Figure 10. Now Work P R O B L E M 3 5 M = 1 - 1, 3 2 x = x 1 + x 2 2 = - 5 + 3 2 = - 1 and y = y 1 + y 2 2 = 5 + 1 2 = 3 1 x , y 2 y 2 = 1 x 1 = - 5, y 1 = 5, x 2 = 3, x y 5 5 –5 P 2 (3, 1) P 1 (–5, 5) M (–1, 3) Figure 10 7. If are the coordinates of a point P in the xy -plane, then x is called the of P and y is the of P. 8. The coordinate axes divide the xy -plane into four sections called . 9. If three distinct points P , Q , and R all lie on a line and if then Q is called the of the line segment from P to R . d 1 P , Q 2 = d 1 Q , R 2 , 1 x , y 2 10. True or False The distance between two points is some- times a negative number. 11. True or False The point lies in quadrant IV of the Cartesian plane. 12. True or False The midpoint of a line segment is found by averaging the x -coordinates and averaging the y -coordinates of the endpoints. 1 - 1, 4 2 Concepts and Vocabulary ‘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. On the real number line the origin is assigned the number . (p.A4) 2. If and 5 are the coordinates of two points on the real number line, the distance between these points is . (pp.A5–A6) 3. If 3 and 4 are the legs of a right triangle, the hypotenuse is . (p.A14) 4. 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) - 3 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) 1.1 Assess Your Understanding In Problems 13 and 14, plot each point in the xy-plane.Tell in which quadrant or on what coordinate axis each point lies. 13. (a) (b) (c) C = 1 - 2, - 2 2 B = 1 6, 0 2 A = 1 - 3, 2 2 14. (a) (b) (c) C = 1 - 3, 4 2 B = 1 - 3, - 4 2 A = 1 1, 4 2 Skill Building (d) (e) (f) F = 1 6, - 3 2 E = 1 0, - 3 2 D = 1 6, 5 2 (d) (e) (f) F = 1 - 3, 0 2 E = 1 0, 1 2 D = 1 4, 1 2 In Words To find the midpoint of a line segment, average the x -coordinates and average the y -coordinates of the endpoints.
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SECTION 1.1 The Distance and Midpoint Formulas 7
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