6.7: Coordinate Proofs (x1 , y1) (x2 , y2) 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)

Examine trapezoid TRAP Examine trapezoid TRAP. Explain why you can assign the same y-coordinate to points R and A. 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. 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

midpoint = midpoint formula Use coordinate geometry to prove that the quadrilateral formed by connecting the midpoints of rhombus ABCD is a rectangle. midpoint = midpoint formula The quadrilateral XYZW formed by connecting the midpoints of ABCD is shown below. 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

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

Coordinate Proofs Prove the midsegment of a trapezoid is parallel to the base. (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!

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

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

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!

Homework 6.7 Remember the honor code. No Copying! 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 Saint Agnes Academy Text Resource: Prentice Hall

Check in INK! Pages 333-337 Exercises 1. a. W , ; Z , b. W(a, b); GEOMETRY LESSON 6-7 Check in INK! Pages 333-337 Exercises 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. c. y-axis a 2 b c + e d 6-7

Check in INK! 5. (continued) d. midpts. e. sides f. slopes GEOMETRY LESSON 6-7 Check in INK! 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 6-7

Check in INK! 10. (continued) GEOMETRY LESSON 6-7 Check in INK! 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 6-7

Check in INK! 17. no; may need measures GEOMETRY LESSON 6-7 Check in INK! 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