Find the unknown coordinates of the point(s) in the figure. Then show that the given statement is true.
17. n ABC > n DEC 18. } PT > }
SR
B(?, ?)
A(2h, 0)
C(0, k)
D(h, 2k)
E(h, k)
S(?, ?)
T (?, ?)
P(0, 0) R(2h, 0)
(h, k)
19. The coordinates of n ABC are A(0, 5), B(8, 20), and C(0, 26). Find the length of each side and the perimeter of n ABC. Then find the perimeter of the triangle formed by connecting the three midsegments of n ABC.
20. Swing Set You are assembling the frame
leg leg
crossbar
?
for a swing set. The horizontal crossbars in the kit you purchased are each 36 inches long. You attach the crossbars at the midpoints of the legs. At each end of the frame, how far apart will the bottoms of the legs be when the frame is assembled? Explain.
21. A-Frame House In an A-frame house,
K
M
PN
L
J
the floor of the second level, labeled } LM , is closer to the first floor, } NP , than is the midsegment } JK . If } JK is 14 feet long, can } LM be 12 feet long? 14 feet long? 20 feet long? 24 feet long? 30 feet long? Explain.
Practice B continuedFor use with the lesson “Midsegment Theorem and Coordinate Proof”
Lesson Midsegment Theorem and Coordinate Proof Teaching Guide
1. (0, 4) 2. (22, 0) 3. (2, 22) 4. (4, 2)
5. A(0, 4), B(22, 0), C(2, 22), D(4, 2); a square
Practice Level A
1. 8 2. 14 3. 4 4. 9 5. } DF 6. } JF , }
GK
7. } KJ 8. } DK , } KF 9. 7 10. 7
11. S(3, 3), T(5, 1), U(2, 0)
12. slope of }
ST 5 21; slope of } PR 5 21; length of
} ST 5 2 Ï
}
2 , length of } PR 5 4 Ï}
2
13. Sample answer: 14. Sample answer:
x
y
1
1
(0, 0) (7, 0)
(7, 4)(0, 4)
x
y
1
1
(0, 0) (s, 0)
(s, s)(0, s)
15. Sample answer: 16. Sample answer:
x
y
2
8 (0, 0)(9, 0)
(0, 5)
x
y
2
8 (0, 0) (14, 0)
(0, 14)
17. Use the Distance Formula to show that }
AB > }
AC .
Practice Level B
1. 14 2. 8 3. 17 4. } JL 5. } JK 6. } RT
7. }
KS , } RT 8. } KR , }
ST 9. }
LT , }
RS
10. Sample answer: (0, 0), (5, 0), (0, 3)
11. Sample answer: (0, 0), (7, 0), (7, 4), (0, 4)
12. Sample answer: (0, 0), (6, 0), (6, 6), (0, 6)
13. Sample answer: (0, 0), (12, 0), (0, 12)
14. 17 15. 37 16. 46 17. B(2h, k); Using the distance formula, AB 5 DE 5 k, BC 5 EC 5 h, and AC 5 DC 5
Ï}
(k 2 1 h 2) . That means }
AB ù } DE , }
BC ù }
EC , }
AC ù
} DC by the definition of congruence. So, n ABC
ù n DEC by the SSS Congruence Postulate.
18. S 1 h } 2 ,
k }
2 2 , T 1 3h
} 2 ,
k }
2 2 ; Using the distance formula,
PT 5 SR 5 Ï}
(9h 2 1 k 2)
} 4 . So, } PT ù
} SR by the
definition of congruence. 19. AB 5 17, BC 5 10, AC 5 21; 48; 24
20. 72 in.; The crossbar is the midsegment of the legs. 21. no; no; yes; yes; no; 14 < LM < 28
Practice Level C
1. } WY 2. } WX 3. 4 4. 14 5. 32 6. 27
7. A(2h, 4k), C(22h, 4k), D(2h, 0)
8. B 1 h } 2 ,
3k }
2 2 , D(h, k), F 1 h }
2 ,
k }
2 2
9. BD is not parallel to }
AD .
10. Use the Midpoint Formula to find the
coordinates of G 1 e } 2 ,
d }
2 2 and H 1 f 2 e
} 2 ,
d }
2 2 .
Use the Distance Formula to show
GH 5 Ï}}
1 f 2 e }
2 2
e } 2 2 2 1 1 d } 2 2
d } 2 2
2 5
f }
2 and
DF 5 Ï}}
( f 2 0)2 1 (0 2 0)2 5 f. So, GH 5 1 }
2 DF.
11. By the definition of an angle bisector, ∠ ABD ù ∠ CBD. Use the Distance Formula to
show AB 5 Ï}
c2
} 4 1 b2 and BC 5 Ï
}
c2
} 4 1 b2 .
Because } BD ù } BD , you can apply the SAS Congruence Postulate to conclude that n ABD ù n CBD. 12. 15; By the Midsegment Theorem, the midsegment of the truss is half of the base of the truss.
Study Guide
1. MN 5 53 cm, ZY 5 14 cm 2. DF 5 7 in.,
BC 5 12 in. 3. }
ST i } BA , ST 5 1 }
2 BA 4. (0, k)
Real-Life Application
1. 4 ft 2. 5 ft 3. yes; Corresponding Angles Converse 4. 6 ft 5. 2 ft
Challenge Practice
1. AB 5 BC 5 7 in., AC 5 10 in. or AB 5 BC 5 10 in., AC 5 4 in.
