Download Math 70 Exam 2 review Use only a compass and straightedge to

Math 70 Exam 2 review
Use only a compass and straightedge to complete the following constructions.
1. Duplicate the three line segments shown below. Label them as they are labeled in the figure.
A
B
C
D
E
F
2. Construct an angle congruent to P.
P
3. Duplicate ABC below.
B
A
C
4. Construct a perpendicular bisector that is congruent to segment r.
r
5. Construct perpendicular bisectors to divide segment s into four congruent segments.
s
6. Construct the median GA in FGH.
F
G
H
7. Construct the midsegment AB in FGH where A is the midpoint of FG and B is the midpoint of GH.
F
G
H
8. Construct a perpendicular from AB that passes through P.
P
A
B
9. Construct the line perpendicular to line p at point Q.
Q
p
10. Name an altitude for BCD.
B
E
D
F
G
C
11. The diagram below shows an obtuse isosceles triangle FGH.
a. Construct the altitude GI to side F H .
b. How could the construction be checked to make sure it was an altitude?
G
F
H
12. Use only a compass and straightedge to complete the following constructions.
Construct the angle bisector of P.
P
13. Is AD an angle bisector of BAC ?
B
10 ft
D
9 ft
A
C
Use only a compass and straightedge to complete the following constructions.
14. Use perpendicular lines to construct a line that passes through point P and is parallel to line l.
l
P
15. Construct a rhombus with x as the length of each side and A as one of the acute angles.
x
A
16. Construct a kite using the segments with lengths x and y as sides.
x
y
17. Construct a rhombus with sides of length p.
p
18. Does the given information determine a specific rhombus or could the constructed rhombus vary? If the
constructed rhombus could vary, how will it vary?
Using only a compass and straightedge, construct a rhombus with a 45-degree angle and sides of length DE.
D
E
[A] The lengths of the sides of the rhombus will vary.
[B] Both the angles and the lengths of the sides of the rhombus will vary.
[C] The angles of the rhombus will vary.
[D] A specific rhombus is determined.
19. Construct the incenter of DFG.
F
D
G
20. Construct the circumcenter of JKL.
K
L
J
21. If point P is the circumcenter of ABC and the length of segment AP is 11 inches, what is the length of
segment CP ?
22. Is point P an incenter? Explain why or why not.
B
J
K
11
P
10
10
A
C
L
23. Is point P a circumcenter? Explain why or why not.
B
7 in.
P
7 in.
7 in.
A
C
24. The circumcenter of a triangle is ___
[A] equidistant from only two of the vertices.
[B] equidistant from only two of the sides.
[C] equidistant from the sides.
[D] equidistant from the vertices.
25. Construct a circle circumscribing the triangle.
26. Construct a circle inscribed in the triangle.
27. Julian wishes to center a butcher-block table in a kitchen’s work triangle, that is, at a location equidistant from
the refrigerator, stove, and sink. Which point of concurrency does Julian need to locate?
28. The first-aid center of Mt. Thermopolis State Park needs to be at a point that is equidistant from three bike
paths that intersect to form a triangle. At what point of concurrency should they locate the center so that in an
emergency medical personnel will be able to get to any one of the paths by the shortest route possible?
[A] circumcenter [B] orthocenter [C] incenter
29. Use only a compass and straightedge to complete the following constructions.
Construct the centroid of JKL.
L
J
K
30. The centroid of a triangle divides each median into two parts so that the distance from the centroid to the
vertex is _____ the distance from the centroid to the midpoint of the opposite side.
31. Point P is the centroid of the triangle shown below. If BL is 27 cm, what is PL?
B
K
J
P
C
L
A
[A] 27 cm
[B] 81 cm
[C] 13.5 cm
[D] 9 cm
32. Point P is the centroid of the triangle shown below. If BP is 22 inches, what is BL?
B
J
K
P
A
[A] 242 in.
C
L
[B] 33 in.
[C] 44 in.
[D] 11 in.
Find the missing measurements.
33. a = _____
35. d = _____
e = _____
f = _____
34. b = _____
36. j = _____
Find the missing measurements.
k = _____
l = _____
Tell whether it is possible to draw a triangle with the given side lengths.
37. 3 in., 4 in., 5 in.
38. 1 cm, 7 cm, 8 cm
39. 3 ft, 5 ft, 9 ft
Arrange the letters in order from greatest value to least value.
40.
41.
42.
43. Find the value of x.
N
84°
x
121°
O
M
44. Refer to the figure and information given below. Give a congruence statement for two triangles in the figure
and name the congruence conjecture that supports the congruence.
H
G
J
I
GJ  JI
45. AC  DC and BC  CE . Write a paragraph proof to show that ABC  DEC.
B
D
C
A
E
46. Determine what information you would need to know in order to use the SSS Congruence Conjecture to show
that the triangles are congruent.
B
A
C
D
[A] BAD  CDB
[B] AD  BD
[C] ADB  CBD
[D] AD  CB
47. Determine which triangles in the figure of a triangle are congruent by SAA.
A
B
C
F
D
E
[B] ADE  EBA [C] ABF  AFE [D] EFD  AFE
[A]  ABF   EDF
48. In the figure below, LJ bisects IJK and ILJ  JLK. Find a congruence statement for the two triangles
in the figure and name the congruence shortcut used.
I
L
J
K
[A] ILJ  KLJ; ASA [B] ILJ  KLJ ; SAA
[C] KLJ  LIJ ; SAA
[D] KLJ  LIJ ; ASA
49. If GHI   JKL, what six things can you conclude about the corresponding parts of the two triangles?
50. Given: BD bisects AC , AB  BC
Show: CBD   ABD
B
A
D
C
51. Find the value of x so that ABC  XYZ.
b
g
b
g
mA  46 , BC  7 x  3 meters, mX  46 , AC  31 meters, YZ  6x  6 meters
B
[A] x  6
A
Z
C
X
[B] x  5
Y
[C] x  3
[D] x  4
52. Use the given information to make a sketch of ABC and DEF. Mark the triangles with the given
information.
B  E
C  F
AB  DE
53. Given: ABED
AB ED 
YY
Show:
BC  DC
A
D
C
B
E
54. Which set of statements and reasons would correctly complete the flow chart proof?
Given: AB  ED
AB ED

Show: BC  DC
YY
A
D
C
B
E
AB  ED
Given
AB || ED
Given
ABC  EDC
AIA
2) _________
BC  DC
CPCTC
1) __________
From the information given, determine which triangles, if any, are congruent. State the congruence conjecture that supports
the congruence statement. If the triangles cannot be shown to be congruent from the information given, write “Cannot be
determined.”
55. MXD  ? Why?
56. BNG   ? Why?
57.NMT =  ? Why?
58.HOW   ? Why?
59. Provide each missing reason or statement in the proof.
Given:
D  C
sDE sEC
Show:
sAE  sBE
Flow-chart Proof:
GIVEN
GIVEN
60. Provide each missing reason or statement in the proof.
Given:
Show:
sACsBC,
sAD  sBE
DCA BCE
Flow-chart Proof:
GIVEN
BCE   _______
GIVEN
Math 70 Exam 2 review ANSWERS
[1]
A
B
C
[6]
D
E
F
A
F
[2]
G
H
[7]
F
[3]
A
B
G
H
[4]
[8]
P
r
A
B
[5]
s
[9]
A
Q
B
p
[15]
[10] BG
[11] a.
G
F
I
H
[16] Sample answer:
b. Using a protractor you could measure one of the
angles at point I and show it is 90°.
[12]
P
[17]
[13] No
[14]
l
P
The angles of the rhombus will vary.
[18] [D]
[19] Sample answer:
[26]
F
D
G
[27] circumcenter
[20] Sample answer:
K
[28] [C]
[29]
L
J
[21] CP  11 in.
[22] No. Point P is not equidistant from each side of [30] twice
the triangle.
[23] Yes. The distance from the circumcenter to the
vertices is equidistant.
[24] [D]
[25]
[31] [D]
[32] [B]
[33] a  80
[34] b  50
[35] d  30 , e  30 , f  14 cm
[36] j  45 , k  45 , l  6 ft
[37] Yes
[38] No
C
F
[39] No
[40] a, b, c
[41] c, b, a
[42] c, b, a
A
B
AB ED
Given
YY
AB  ED
Given
ABCEDC
AIA Conj.
ACBECD
VA Conj.
[45]  ACB   DCE because they are vertical
angles. So ABC  DEC by SAS.
[54] 1) ACB  ECD; VA Conj.
2) ACB  ECD; SAA
[46] [D]
[47] [A]
[55] ∆𝑀𝑌𝑇 , 𝑆𝐴𝑆
[48] [A]
[49] GH  J K , H I  K L, GI  J L,
G   J ,  H  K,  I   L
[50] We are given that BD bisects AC and AB  BC.
The definition of bisect tells us that AD  CD, and
BD  BD because they are the same segment. The SSS
shortcut gives ADB  CDB, so CBD   ABD by
CPCTC.
[51] [C]
[52] Sample answer:
E
[53]
[43] 155°
[44] GHJ  IHJ by the SAS Conjecture
D
[56] CANNOT BE DETERMINED
[57] CANNOT BE DETERMINED
[58] ∆𝑇𝑌𝑊 , 𝐴𝑆𝐴
[59] ASK
[55] ASK
ACBECD
SAA