Geometry Semester 1 Review
Name:
Right Triangles – Basic
1. Find the value of each variable below.
2. Find the length of the line segment joining the points (-2,3) and (4,-2).
Right Triangles – Medium
3. Tyree planned to build a ramp from his window to the sidewalk so he could skateboard right
out his window on his way to school. His window is 5 feet off the ground and the sidewalk is
20 feet from his house.
a.
Draw a diagram to represent this situation.
b.
How long will Tyree's ramp need to be to make it from his window to the sidewalk?
c.
He knows that if the angle the ramp makes with the ground is more than 20 he will gain too
much speed and lose control when he hits the sidewalk. What is the angle his ramp will make with the
ground? Is his ramp plan safe?
4. Markwick Farble, that great trig student, now has his sights set on the school's flagpole. If
our 6' tall hero is 30' from the pole and sights its top at a 40
flagpole?
angle, how high is the
Right Triangles – Advanced
5. Graph and label the points A(1, 1), B(5, 1), and C(4, 3).
Draw
.
a. Find
b.
.
What is the perimeter of the triangle?
6. Find the value of x in the diagram below.
Transformations – Basic
7. Translate down 4, right 3
Rotate 180o around origin
Reflect over y-axis.
Transformations – Medium
8. Rotate 90o clockwise around origin
Reflect over y=x.
9. For each of the following, draw in all possible lines of symmetry. If a shape does not have
any lines of symmetry, explain why.
Transformations – Advanced
10. Rosalie and Emmet are having a disagreement. Rosalie thinks a line can have a line of
symmetry, while Emmet disagrees. What do you think: can a line have a line of symmetry?
Why? Explain completely.
11. Identify the transformations used on triangle ABC below.
Angles – Basic
12. Find the measure of <ABC below.
13. Find the values of x and y.
14. Find the value of x.
Angles – Medium
15. If z = 40o, find the measures of x and y.
16. Find the value of x and all the missing angles.
Angles – Advanced
a.
b.
c.
d.
e.
f.
g.
h.
17. Name the relationship between each pair of
angles.
a,h,g
d and e
b and c
g and d
g and f
c and f
d and f
Now find all the missing angle measures.
18. If <3 = 50o and <12 = 65o, find all the missing
angle measures below.
19. Find the value of x, y, and z.
2D/3D – Basic
20. For each diagram draw a height to the labeled base and label it 6” long. Find the area of the
figures.
21. Find the area of the shape below.
22. Find the area of the trapezoid.
2D/3D – Medium
2
23. If the area of ΔABC is 30 in , what is the length of KC?
24. Find the area of each figure below.
25.
26.
2D/3D – Advanced
27. Find the length of one side of the square shown below.
28. Plot the points (-2,4) (3,1) (5,-2) and (-1,-1) on the grid below. Connect them and find the area
of the shape.
29. Find the area of each shape below.
30. Find the area of each shape below.
Probability – Basic
31. My mother will call every day between 4:30 pm and 7:00 pm. If we eat dinner from 6:00 to
6:30 each night, what is the probability that my mother will interrupt our meal?
32. A bag contains 31 triangles. I choose one at random. What is the probability that the
triangle I choose has two obtuse angles? Explain.
33. When Mark and Courtney were finished categorizing all their shapes from their Shape
Bucket, Mark quickly threw all the shapes
back into the empty bucket. All that was left
was their Venn diagram showing how many
shapes were listed in each category.
“Wait!” Courtney exclaimed. “We weren’t
finished yet!”
“We have our Venn diagram finished. What else is there?” Mark replied.
“We need one of the quadrilaterals for the next problem. Reach in and grab one.”
If Mark reaches in and chooses a shape from this collection in the Shape Bucket, what is the
probability the shape he chooses will be a quadrilateral?
34. The bar graph at right shows the probability of different colored
sections coming up when a spinner is spun. Unfortunately, part of
the graph has been ripped off. If only one color is missing, green,
from this picture, what is the probability that green will come up?
Explain completely, justifying your answer.
35. Alex is trying to stump Hailey. He has the told her that he has a spinner just like the
hexagon shown at right, with three colors on it: red, yellow and blue. He will spin it 100
times, and tell her the outcome each time. Based on the responses, Hailey has to figure out
how the spinner is colored. She keeps track and finds these results:
Red: 49
Yellow: 36
Blue: 15
How should Hailey color in the hexagon? Justify your answer completely.
Probability – Medium
36.
37.
38. At a new ice cream store, you spin the two spinners below to determine what ice cream you
will be receiving. If you will only be happy if you get chocolate or a cone, what is the
probability that you are happy with what you get?
39. Janet had trouble choosing outfits, so she made spinners to randomly select outfits. Based
on her spinners below, what is the probability that she wears a hat or scarf with boots?
40. When Ronald goes to BIG Burgers, he orders a sandwich and a side order. For the sandwich,
50% of the time he orders a Jumbo Burger, 30% of the time he orders a Chicken Sandwich
and 20% of the time he orders a Regular Burger. For the side, 25% of the time he orders
Jumbo Fries, 40% of the time he orders Curly Fries and the rest of the time he orders a
salad. Ronald just bought lunch. What is the probability that he ordered a Jumbo Burger
with Regular Fries or a Chicken Sandwich with a Salad? Make an area or tree diagram to help
with the problem.
41. The two spinners shown to the right are spun. If the letters match, you win. Otherwise you
lose. What is the probability that you win? Use an area or tree diagram to help solve the
problem. Show all calculations that lead to your results.
42. At her archery lessons, Tess has a 90% chance of getting a bull’s eye. Unfortunately, there is
only a 40% chance that her instructor will be watching her when she shoots. What is the
probability that her instructor will see Tess hit the bull’s eye? Explain completely.
Probability – Advanced
43. Boris the rat is trying to find food in the maze shown below. Boris is equally likely to take
any path when coming to an intersection. If the food is stored in area A, what is the
probability that Boris finds food? Show all of your calculations!
44. Eddie Hascal heard through the grapevine that the quiz on probability had only three
questions. He also heard that one question had an answer of zero, another had an answer of
one, and a third had an answer of one-half. Eddie did not study at all for this quiz, so his
plan is to randomly choose an answer from these three for each question, without repeating
an answer. If the information Eddie heard is correct, what is the probability that Eddie will
score a perfect on the quiz? Explain completely.
45. Less is known for making errors when correcting his students’ papers. On any given problem,
he has a 40% chance of grading the problem correctly. (That is, marking a correct solution as
correct or a wrong solution as incorrect.)
Victoria is pretty good at math, but on any given problem she has an 80% chance of getting the
correct answer. Victoria just took a one-question quiz in Mr. Less’ class. What is the probability
that she will get 100%? What is the probability that she does the problem correctly, but is
marked as wrong? Explain completely.
46. The accuracy of polygraph tests has been questioned for many years. Some studies put the
accuracy at 80%. Estimates have most people lying 15% of the time. Given this information,
what is the probability that someone is telling the truth, but the polygraph says the person
is lying (a false positive)?
Algebra Review – Basic
47. Write an equation of a line with a slope of 2/3 whose graph will cross the y-axis at the point
(0, 12.65).
48. a.
b.
Solve for x and check your solution: 5x + 6 = 31
Simplify: 4x2 – 10x – 3x2 + 2x + 1
49. Solve each of the following:
12 − 3𝑥 = −2(4 − 𝑥)
2𝑥 − (4 − 3𝑥) = 12 + 3(𝑥 − 5)
50. Solve the system:
3x + 2y = –23
y=x–4
Algebra Review – Medium
51. A line goes through the points (–10, –23) and (20, 37). Another point on the line is
A.
(2, 19)
B.
(2, –3)
C.
(2, –53)
D.
(2, 1)
E.
None of These
52. Solve the following equations for x.
𝑥+
53. a.
b.
1−𝑥
2
=
𝑥+3
4
−
2
42
3
𝑥
=
32
12
21
𝑥
Find the equation of the line graphed.
Graph the line 2x – 3y = 12 on the same grid.
54. Solve for x:
2
5
𝑥−
1
10
3
= 𝑥
4
=
𝑥
𝑥+4
3
6
=
9
20
55. Given the system below, what is the value of a + b?
3a + 2b = 11
a–b=7
A.
2
B.
2.2
C.
3
D.
7
E.
None of these
56. Solve for x and y in the system below:
2x – y
x + 3y = –9
57. Solve for x and y:
5x + 2y = 10
5x + 2y = 20
58. Write the equation of the linear function passing through the points (3, 7) and (5, 12).
Algebra Review – Advanced
59. Write an equation for each line segment in the triangle at
right. Describe any relationships that you notice about line
segments and the equations.
60. True or false: (x + y)2 = x2 + y2
61. Factor: x2 + 3x – 28
62. Solve the equation below for q. Check your answer(s) if possible.
63. Write in simple radical form:
64. Write in simple radical form:
65. Write in simple radical form:
66. Find the value of x in the triangle below.
67. Solve for x:
−2
𝑥−4
=
4
1−𝑥
68. Solve the system of equations.
1
𝑥−𝑦=1
2
1
2
𝑥− 𝑦=2
3
3
Coordinate Geometry – Basic
69. Graph the equation
5
𝑦 = 2 𝑥 − 4.
a.
Graph the line that is perpendicular to this
line and goes through the point (1,-2).
b.
Write the equation for the line you drew in part
(a) of this problem.
Coordinate Geometry – Medium
70. Find the equation of the line perpendicular to
71. Find the equation of the line parallel to
that goes through (6, –2).
, that goes through (–4, 1)
Coordinate Geometry – Advanced
72. A line goes through the points (5, –1) and (7, –5). A line perpendicular to that line is
A.
B.
C.
D.
E.
None of these
73. Graph the equations below on the same set of axes. When
the lines intersect, do they form an acute angle, a right
angle, or an obtuse angle? Explain how you know. Be
complete and clear.
74. In the figure at right, if ∆ABC is rotated about the origin 90°, how will the slopes of the
sides of the triangle change? Be clear and complete, justifying your statements.
Building Blocks – Basic
75. If the perimeter of the rectangle below is 84 inches, find the value of x.
76. Draw an equilateral triangle. If its sides are 8 inches long, what is the area of the triangle?
77. The perimeter of triangle ABC is 56 cm. Find the length of AB.
78. Hugo and Rose love to collect triangles. Hugo has a collection of ninety-eight 30o-60o-90o triangles,
of which he is very proud, thank you very much. Rose is partial to isosceles right triangles, and her
collection contains 102 of them.
a.
If Hugo reaches into his bag of 30 -60 -90 triangles and pulls out two triangles, what is the
probability that the two triangles are similar? Explain your answer completely.
b.
If Rose reaches into her bag of isosceles right triangles and pulls out two triangles, what is the
probability that the two triangles are similar? Explain your answer completely.
c.
If Hugo pulls out one triangle from his bag, and Rose pulls out one triangle from her bag, what is
the probability that the two triangles are similar? Explain completely.
Building Blocks – Medium
79. Find the area and perimeter of the shape below.
80. Find the area and perimeter of the shaded region.
81. Isosceles
ABC has two equal sides of 10 cm and base angles of 50 . Find the perimeter
of the triangle.
82. Willis has just finished sorting all of his triangles in his collection into several boxes. The
blue box contains only equilateral triangles. The green box contains only scalene triangles.
The red box contains only isosceles right triangles, while the yellow box contains isosceles
triangles that are not right. If Willis is to choose two triangles at random from each box,
what is the probability the two chosen triangles are similar? Note: you need to answer for
each box, and you may have to give a reasonable estimate for some.
Building Blocks – Advanced
83. Find the area and perimeter of the shape below.
84. An acute angle of a parallelogram measures 50 . If a base measures 15 centimeters and the
other side measures 10 centimeters, what is the area of the parallelogram?
85. Find the perimeter and area of the figure below.
86. In a blue box are the numbers 3 and 5, in a red box are the numbers 4, 5 and 12, while in a
green box are the numbers 5 and 13. We will be trying to make triangles with the 3 numbers
drawn.
a.
What is the probability that a triangle can be formed?
b.
What is the probability that an isosceles triangle can be formed?
c.
What is the probability that a equilateral triangle can be formed ?
d.
What is the probability that a right triangle can be formed?
87. Find the area and perimeter of the trapezoid below.
88. Solve for x.
Proves Theorems – Basic
89. Determine if each pair of triangles below is similar. If so, draw a flowchart. If not, explain
why not.
Proves Theorems – Medium
90.
Proves Theorems – Advanced
91. Draw two triangles that are similar by each of the four shortcuts: SSS~, SAS~, AA~, and HL~.
Similarity – Basic
92.
93.
94. Explain what it means for two figures to be similar.
95. Given
, which of the following
could be used to solve for x ? Note: the picture is not
drawn to scale.
𝑥
4
=6
11
𝑥
4
=6
12
𝑥
6
= 11
4
𝑥
11
= 12
4
none of these
Similarity – Medium
96.
97.
98. True or false: If two figures are congruent (identical) then they are also similar. Explain.
99. True or false: If two figures are similar, then they are also congruent (identical). Explain.
100.
True or false: All isosceles triangles are similar. Explain.
Similarity – Advanced
101.
102.
103.
Name one type of shape that is always similar.
Name another type of shape that is always similar.
104.
In the figure at right, BD//AE.
a.
Are
and
not? Explain completely.
similar? Why or why
b.
If BC = 6, AB = 8, and AE = 9, what is the length of BD?
Show all work.
105.
Find the value of x.