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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