Download Answers for the lesson “Midsegment Theorem and Coordinate Proof”

LESSON
5.1
Answers for the lesson “Midsegment
Theorem and Coordinate Proof”
Skill Practice
21.
y
B(p, q)
1. midsegment
2. Sample answer: The vertex of
the right triangle is located at the
origin and the other two
vertices are located on the
x-axis and y-axis which limits the
number of variables need to label
them; label the vertex on the
x-axis (2a, 0) and the vertex on
the y-axis (0, 2b).
3. 13
6. }
YZ
9. }
JX, }
KL
4. 10
7. }
XZ
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11. }
YL, }
LZ
A (0, 0)
}
q
p q
AB 5 Ïp2 1 q2 , }p, 1 }2, }2 2;
}
q
1
3q q
2
BC 5 Ïp 2 1 q 2 , 2}p, }
,} ;
2 2
CA 5 2p, 0, ( p, 0); no; yes;
it’s not a right triangle because
none of the slopes are negative
reciprocals and it is isosceles
because two of the sides are the
same length.
5. 6
8. }
KL
10. }
XK, }
KZ
12–19. Sample answers are given.
C (2p, 0) x
22.
y
B (h, h)
12. (0, 0), (3, 0), (0, 2)
13. (0, 0), (7, 0), (0, 7)
14. (0, 0), (3, 0), (3, 3), (0, 3)
A(0, 0)
C (2h, 0) x
15. (0, 0), (2m, 0), (a, b)
16. (0, 0), (a, 0), (a, b), (0, b)
17. (0, 0), (s, 0), (s, s), (0, s)
18. (0, 0), ( p, 0), (0, p)
19. (0, 0), (r, 0), (0, s)
}
20.
Ïr 2 1 s 2 ; yes
AB 5 hÏ2 , 1, 1 }2, }2 2;
}
h h
,} ;
BC 5 hÏ2 , 21, 1 }
2 22
}
3h h
CA 5 2h, 0, (h, 0); yes; yes; it is
a right triangle and an isosceles
triangle since two sides are both
congruent and perpendicular.
Geometry
Answer Transparencies for Checking Homework
134
30. H(2h, k), G(h, k); slope of
y
23.
A(0, n)
k
}
HE 5 2} and the slope of
B(m, n)
3h
k
}
DG 5 }.
3h
31. A
C(m, 0)
x
32.
y
m
AB 5 m, 0, 1 }
,n ;
2 2
U(4, 5)
n
BC 5 n, undefined, 1 m, }2 2;
}
n
V(7, 4)
m n
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, }, } ; yes;
CA 5 Ïm2 1 n2 , 2}
m 1 2 22
no; one side is vertical and
one side is horizontal thus the
triangle is a right triangle. It is
not isosceles since none of the
sides are the same length.
24. 14
25. 13
26. 34
27. You don’t know that }
DE and }
BC
are parallel.
28. PQ 5 13, PR 5 14, QR 5 15,
perimeter 5 42; 84
T (2, 1)
x
(21, 2), (5, 0), (9, 8)
1
1
33. GE 5 }DB, EF 5 }BC,
2
2
1 1
1
}BC
DB
area of nEFG 5 }2 }
2
2
1
5 }8(DB)(BC),
1
area of nBCD 5 }2 (DB)(BC)
F
1
2G
29. (0, k). Sample answer: Since
nOPQ and nRSQ are right
triangles with }
OP > }
RS and
}
}
PQ > SQ, the triangles are
congruent by SAS.
Geometry
Answer Transparencies for Checking Homework
135
34. Find the slope of each
midsegment which gives you the
slope of each side of the triangles.
Use the slope and the known point
on each side to find the equation
of the line containing each side.
Solve the three systems to find the
vertices of the triangle; in
Exercise 32, the vertices were
found graphically and in this
exercise, the vertices were
found algebraically.
Problem Solving
35. 10 ft
}
36. Since PR 5 Ï h 2 1 k 2 and
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}
PQ 5 Ïh 2 1 k 2 , nPQR is
isosceles by definition.
37. The coordinates of W are (3, 3)
and the coordinates of V are
(7, 3). The slope of }
WV is 0 and
}
the slope of OH is 0 making
}
WV i }
OH. WV 5 4 and OH 5 8
40. Definition of midsegment,
Midsegment Theorem and
Corresponding Angles Postulate,
Midsegment Theorem and
Corresponding Angles Postulate,
PST, SQU, ASA
41. Sample answer: The coordinates
of D are (q, r) and the coordinates
of F are ( p, 0) since
2p 1 0
010
} 2 5 ( p, 0). The
,
1}
2
2
slope of }
DF is }
q 2 p and
q2p5}
r20
r
BC is
the slope of }
}}
}5}
q 2 p , so DF i BC.
2r 2 0
2q 2 2p
r
}}
DF 5 Ï (p 2 q)2 1 r 2 and
}}
BC 5 Ï(2q 2 2p)2 1 (2r)2 5
}}
2Ï( p 2 q)2 1 r 2 making
1
DF 5 }2 BC.
1
thus WV 5 }2OH.
38. No, no, yes, no; AB 5 4 feet since
}. The
it is half the length of EF
} must be greater than
length of CD
4 feet but less than 8 feet.
39. 16. Sample answer: }
DE is half
the length of }
FG which makes
}
FG 5 8. FG is half the length of
}
AC which makes AC 5 16.
Geometry
Answer Transparencies for Checking Homework
136
42. Sample answer: Let A(0, 0),
B(0, p), and D(q, 0) be the
vertices of n ABD. Since C is the
midpoint of }
BD, its coordinates
q p
are 1 }2 , }2 2.
Î1 2 1 2
Î1 2 1
5Ï
, and
DC 5 Î1 2 q 2 1 1
}
p 2
p 2 1 q2
q 2
Ï
AC 5 }2 1 }2 5 }
,
2
}}
p 2
q
2
BC 5 }2 2 0 1 p 2 }2
2
}
p2 1 q2
}
2
}}
q
2
2
}
}
p2 1 q2
Ï
5 },
2
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triangle ABC with A(0, p), B(0, 0),
C( p, 0), and right angle B.
AC.
D1 }2 , }2 2 is the midpoint of }
p p
}
}
so AC 5 BC 5 DC.
1
43. a. }
2
5
b. }
4
19
c. }
8
44. Sample answer: Right isosceles
p
2
}20
2
2
pÏ 2
AB 5 p, BD 5 }
,
2
}
pÏ 2
DA 5 }
, CB 5 p,
2
}
pÏ 2
DC 5 }
which makes
2
the triangles congruent by SSS.
In each triangle, pairs of sides
are congruent making each
triangle isosceles. Since ŽCDB
and ŽADB are a linear pair and
are congruent, they are right
angles, which makes the
triangles congruent right isosceles
triangles.
45. Sample answer: nABD and
nCBD are congruent right
isosceles triangles with A(0, p),
p p
B(0, 0), C(p, 0), and D 1 }2, }2 2.
AB is
AB 5 p, BC 5 p, and }
}
vertical and BC is horizontal, so
}
AB > }
BC. By definition, nABC is
a right isosceles triangle.
Geometry
Answer Transparencies for Checking Homework
137
46. Sample answer: A segment
joining the midpoints of }
DL
}
and EN; the length of the
3
quarter-segment will be }4 the
LN and the length of
length of }
7
the eighth-segment will be }8
LN; nLMN,
the length of }
}
midsegment XY, quarter-segment
}
DE, and eighth-segment }
FG.
L(0, 0), M(a, b), and N(c, 0)
a1c b
, }2 2,
leading to X 1 }2, }2 2, Y 1 }
2
a b
a 1 3c b
D 1 }4 , }4 2, E 1 }
, }4 2, F 1 }8, }8 2,
4
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a b
a b
a 1 7c b
and G1 }
, }8 2. LN 5 c,
8
c
3
7
XY 5 }2, DE 5 }4 c, and FG 5 }8c.
Geometry
Answer Transparencies for Checking Homework
138