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6.7: Coordinate Proofs

With these formulas you can use coordinate geometry to prove theorems that address length (congruence / equality / mid point) and slope ( parallel and perpendicular.). 6.7: Coordinate Proofs. (x 1 , y 1 ). (x 2 , y 2 ). In a trapezoid, only one pair of sides is parallel.

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6.7: Coordinate Proofs

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  1. With these formulas you can use coordinate geometry to prove theorems that address length (congruence / equality / mid point) and slope ( parallel and perpendicular.) 6.7:Coordinate Proofs (x1 , y1) (x2 , y2)

  2. In a trapezoid, only one pair of sides is parallel. In TRAP, TP || RA . Because TP lies on the horizontal x-axis, RA also must be horizontal. Examine trapezoid TRAP. Explain why you can assign the same y-coordinate to points R and A. The y-coordinates of all points on a horizontal line are the same, so points R and A have the same y-coordinates. 6-7

  3. The quadrilateral XYZW formed by connecting the midpoints of ABCD is shown below. Use coordinate geometry to prove that the quadrilateral formed by connecting the midpoints of rhombus ABCD is a rectangle. midpoint = midpoint formula From Lesson 6-6, you know that XYZW is a parallelogram. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle by Theorem 6-14. congruent = distance formula 6-7

  4. XZ = (–a – a)2 + (b – (–b))2 = ( –2a)2 + (2b)2 = 4a2 + 4b2 YW = (–a – a)2 + (– b – b)2 = ( –2a)2 + (–2b)2 = 4a2 + 4b2 (continued) Because the diagonals are congruent, parallelogram XYZW is a rectangle. 6-7

  5. Prove the midsegment of a trapezoid is parallel to the base. Coordinate Proofs (b,c) (d,c) The bases are horizontal line with a slope equal to zero. Is this true for the midsegment? (0,0) (a,0) Conclusion: The midsegment of a trapezoid is parallel to the two bases!

  6. With some experience, you will begin to see the advantage of using the following coordinates: Coordinate Proofs (2b,2c) (2d,2c) (0,0) (2a,0)

  7. Prove the midsegment of a trapezoid is equal to half the sum of the two bases. Coordinate Proofs (2b,2c) (2d,2c) (0,0) (2a,0) 1/2 (2a+2d-2b) = a + d - b =d + a - b

  8. 2. Prove that the diagonals of a parallelogram bisect each other If the diagonals BISECT, then they will have THE SAME midpoint. (2b,2c) (2b+2a,2c) B C D A E (0,0) (2a,0) Since the diagonals have the same midpoint, they bisect each other!

  9. Saint Agnes Academy Text Resource: Prentice Hall Homework 6.7 Page 333 Due at the beginning of the next class. Name Section # Page # Remember the honor code. No Copying! Show your work here IN PENCIL I pledge that I have neither given nor received aid on this assignment

  10. Pages 333-337 Exercises a 2 b 2 1.a.W , ; Z , b.W(a, b); Z(c + e, d) c.W(2a, 2b); Z(2c + 2e, 2d) d. c; it uses multiples of 2 to name the coordinates of W and Z. 2.a. origin b. x-axis c. 2 d. coordinates 3.a. y-axis b. Distance 4. a. rt. b. legs 4. (continued) c. multiples of 2 d.M e.N f. Midpoint g. Distance 5.a. isos. b.x-axis c.y-axis d 2 c + e 2 Check in INK! GEOMETRY LESSON 6-7 6-7

  11. 5. (continued) d. midpts. e. sides f. slopes g. the Distance Formula 6. a. (b + a)2 + c2 b. (a + b)2 + c2 7.a. a2 + b2 b. 2 a2 + b2 8. a. D(–a – b, c), E(0, 2c), F(a + b, c), G(0, 0) b. (a + b)2 + c2 c. (a + b)2 + c2 d. (a + b)2 + c2 e. (a + b)2 + c2 f. g. 8. (continued) h. – i. – j. sides k. DEFG 9. a. (a, b) b. (a, b) c. the same point 10. Answers may vary. Sample: The c a + b c a + b c a + b c a + b Check in INK! GEOMETRY LESSON 6-7 6-7

  12. 10. (continued) Midsegment Thm.; the segment connecting the midpts. of 2 sides of the is to the 3rd side and half its length; you can use the Midpoint Formula and the Distance Formula to prove the statement directly. 11. a. b. midpts. 11. (continued) c. (–2b, 2c) d. L(b, a + c), M(b, c), N(–b, c), K(–b, a + c) e. 0 f. vertical lines g. h. 12–24. Answers may vary. Samples are given. 12. yes; Dist. Formula 13. yes; same slope 14. yes; prod. of slopes = –1 15. no; may not have intersection pt. 16. no; may need measures Check in INK! GEOMETRY LESSON 6-7 6-7

  13. Check in INK! GEOMETRY LESSON 6-7 12 n 12 n 17. no; may need measures 18. yes; prod. of slopes of sides of A = –1 19. yes; Dist. Formula 20. yes; Dist. Formula, 2 sides = 21. no; may need measures 22. yes; intersection pt. for all 3 segments 23. yes; slope of AB = slope of BC 24. yes; Dist. Formula, AB = BC = CD = AD 25. 1, 4, 7 26. 0, 2, 4, 6, 8 27. –0.8, 0.4, 1.6, 2.8, 4, 5.2, 6.4, 7.6, 8.8 28. –1.76, –1.52, –1.28, . . . , 9.52, 9.76 29. –2 + , –2 + 2 , –2 + 3 , . . . . , –2 +(n – 1) 30. (0, 7.5), (3, 10), (6, 12.5) 31. –1, 6 , 1, 8 , (3, 10), 5, 11 , 7, 13 32. (–1.8, 6), (–0.6, 7), 12 n 12 n 2 3 1 3 2 3 1 3 6-7

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