6.7:Coordinate Proofs

(x1 , y1)

(x2 , y2)

midpo int x1 x22

,y1 y22

slopey2 y1 x2 x1

d (x2 x1)2 (y2 y1)

2

With these formulas you can use coordinate geometry to prove theorems that address length (congruence / equality / mid point) and slope ( parallel and perpendicular.)

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.

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.

6-7

Use coordinate geometry to prove that the quadrilateral formed by connecting the midpoints of rhombus ABCD is a rectangle.

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.

6-7

midpoint = midpoint formula

congruent = distance formula

Because the diagonals are congruent, parallelogram XYZW is a rectangle.

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)

6-7

(0,0) (a,0)

(b,c) (d,c)b

2,c

2

d a2,c

2

Coordinate ProofsProve the midsegment of a trapezoid is parallel to the base.

x2 x12

,y2 y12

The bases are horizontal line with a slope equal to zero.Is this true for the midsegment?m

y2 y1x2 x1

mc2 c2

d a2 b2

m 0

Conclusion:The midsegment of a trapezoid is parallel to the two bases!

(0,0) (2a,0)

(2b,2c)(2d,2c)

Coordinate ProofsWith some experience, you will begin to see the advantage of using the following coordinates:

x2 x12

,y2 y12

2b

2,2c

2

b,c 2d 2a2

,2c

2

2(d a)2

,2c

2

(d a),c

(0,0) (2a,0)

(2b,2c)(2d,2c)

Coordinate ProofsProve the midsegment of a trapezoid is equal to half the sum of the two bases.

d (x2 x1)2 (y2 y1)

2

b,c (d a),c dbottom (2a 0)2 (0 0)2

dbottom (2a)2 2a

dtop (2d 2b)2 (2c 2c)2

dtop (2d 2b)2(2d 2b)

dmid (d a b)2 (c c)2

dmid (d a b)2 d a b

1/2 (2a+2d-2b)= a + d - b=d + a - b

(0,0) (2a,0)

(2b,2c) (2b+2a,2c)

2. Prove that the diagonals of a parallelogram bisect each other

A

B CDE

If the diagonals BISECT, then they will have THE SAME midpoint.

x2 x12

,y2 y12

BDmidpo int 2b 2a2

,2c

2

b a,c

ACmidpo int 2b 2a2

,2c

2

b a,c

Since the diagonals have the same midpoint, they bisect each other!

Homework 6.7Page 333Due at the beginning of the next class.

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Text Resource: Prentice Hall

GEOMETRY LESSON 6-7GEOMETRY LESSON 6-7

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.

b. x-axis

c. y-axis

a2

b2

c + e 2

d2

6-7

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GEOMETRY LESSON 6-7GEOMETRY LESSON 6-7

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

6-7

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GEOMETRY LESSON 6-7GEOMETRY LESSON 6-7

6-7

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

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GEOMETRY LESSON 6-7GEOMETRY LESSON 6-7

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

12 n

12 n

23

13

23

13

6-7

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