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Geometry
Pythagorean theorem
A piano mover uses a ramp to move a piano into a house. The doorway to the house is
2
2
2
feet above the ground and the ramp starts
7
7
7
feet from the doorway. Assuming the ground is level and is perpendicular to the side of the house, what is the approximate length of ramp? You must round your answer to two decimal places.
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25
25
25
. A tree
32
32
32
meter-high is broken off
7
7
7
meter from the ground. How far from the foot of the tree will the top strike the ground?
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Point
N
N
N
is directly above point
M
M
M
. Point
M
M
M
is
8
ft
8\,\text{ft}
8
ft
below the ground, and point
N
N
N
is
40
ft
40\,\text{ft}
40
ft
above the ground. Find the distance, in
ft
\text{ft}
ft
, between the points?
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15
15
15
. A ship anchored in a port has a ladder which hangs over the side. The length of the ladder is
200
c
m
200 \mathrm{~cm}
200
cm
, the distance between each rung in
20
c
m
20 \mathrm{~cm}
20
cm
and the bottom rung touches the water. The tide rises at a rate of
10
c
m
10 \mathrm{~cm}
10
cm
an hour. When will the water reach the fifth rung?
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Ten lampposts are equally spaced along a straight line. The distance between two consecutive lampposts is
40
40
40
meters. What is the distance between the third and the eight lampposts?
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The following figure is made of triangle and rectangle. A quadrilateral with one pair of parallel sides. The base lengths are
9
9
9
units and
13
13
13
units. The height is
5
5
5
units. A dashed line that is perpendicular to the base creates two triangles, labeled A and B. A quadrilateral with one pair of parallel sides. The base lengths are
9
9
9
units and
13
13
13
units. The height is
5
5
5
units. A dashed line that is perpendicular to the base creates two triangles, labeled A and B. Find the area of each part of the figure and the whole figure.
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Gabriel leans a
18
18
18
-foot ladder against a wall so that it forms an angle of
7
3
∘
73^{\circ}
7
3
∘
with the ground. How high up the wall does the ladder reach? Round your answer to the nearest hundredth of a foot if necessary.
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A boat is heading towards a lighthouse, whose beacon-light is
140
140
140
feet above the water. The boat's crew measures the angle of elevation to the beacon,
1
0
∘
10^{\circ}
1
0
∘
. What is the ship's horizontal distance from the lighthouse (and the shore)? Round your answer to the nearest tenth of a foo necessary.
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An electrician leans an extension ladder against the outside wall of a house so that it reaches an electric box
33
33
33
feet up. The ladder makes an angle of
7
4
∘
74^{\circ}
7
4
∘
with the ground. Find the length of the ladder. Round your answer to the nearest tenth of a foot if necessary.
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From the observation deck of a skyscraper, Brandon measures a
4
5
∘
45^\circ
4
5
∘
angle of depression to a ship in the harbor below. If the observation deck is
1140
1140
1140
feet high, what is the horizontal distance from the base of the skyscraper out to the ship? Round your answer to the nearest tenth of a foot if necessary.
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From his eye, which stands
1.63
1.63
1.63
meters above the ground, Isaac measures the angle of elevation to the top of a prominent skyscraper to be
1
7
∘
17^\circ
1
7
∘
. If he is standing at a horizontal distance of
294
294
294
meters from the base of the skyscraper, what is the height of the skyscraper? Round your answer to the nearest hundredth of a meter if necessary.
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From a hot-air balloon, Violet measures a
2
4
∘
24^\circ
2
4
∘
angle of depression to a landmark that’s
806
806
806
feet away, measuring horizontally. What’s the balloon’s vertical distance above the ground? Round your answer to the nearest tenth of a foot if necessary.
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An electrician leans an extension ladder against the outside wall of a house so that it reaches an electric box
29
29
29
feet up. The ladder makes an angle of
7
1
∘
71^\circ
7
1
∘
with the ground. Find the length of the ladder. Round your answer to the nearest tenth of a foot if necessary.
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A
20
20
20
-foot fall flagpole is leaning towards a house. If the flagpole makes an
8
5
∘
85^\circ
8
5
∘
angle with the ground and the angle of elevation from the base of the house to the top of the pole is
55
55
55
degrees, find the distance from the base of the flagpole to the house.
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A train is moving with a speed of
60
KM/H
60\,\text{KM/H}
60
KM/H
, how much time would it take to cross a bridge of
250
meter
250\,\text{meter}
250
meter
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Roselyn is driving to visit her family, who live
150
kilometers
150 \text{ kilometers}
150
kilometers
away. Her average speed is
60
kilometers per hour
60 \text{ kilometers per hour}
60
kilometers per hour
. The car's tank has
20
liters
20 \text{ liters}
20
liters
of fuel at the beginning of the drive, and its fuel efficiency is
6
kilometers per liter
6 \text{ kilometers per liter}
6
kilometers per liter
. Fuel costs
0.60
dollars per liter
0.60 \text{ dollars per liter}
0.60
dollars per liter
.
\newline
How long can Roselyn drive before she runs out of fuel?
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Chelsea is sitting
8
8
8
feet from the foot of a tree. From where she is sitting, the angle of elevation of her line of sight to the top of the tree is
3
6
∘
36^{\circ}
3
6
∘
. If her line of sight starts
1.5
1.5
1.5
feet above ground, how tall is the tree, to the nearest foot?
\newline
(1)
8
8
8
\newline
(2)
7
7
7
\newline
(3)
6
6
6
\newline
(4)
4
4
4
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Chelsea is sitting
8
8
8
feet from the foot of a tree. From where she is sitting, the angle of elevation of her line of sight to the top of the tree is
3
6
∘
36^{\circ}
3
6
∘
. If her line of sight starts
1.5
1.5
1.5
feet above ground, how tall is the tree, to the nearest foot?
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19
19
19
. Ms. Herbert has two rectangular bulletin boards in her classroom. Both bulletin boards have the same perimeter, but different areas. The first bulletin board is shown below. What could be the dimensions of the second bulletin board?
\newline
C.
\newline
8
ft
×
12
ft
8\text{ft}\times12\text{ft}
8
ft
×
12
ft
\newline
D.
\newline
10
ft
×
12
ft
10\text{ft}\times12\text{ft}
10
ft
×
12
ft
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A forest ranger sights a fire directly to the south. A second ranger,
7
7
7
miles east of the first ranger, also sights the fire. The bearing from the second ranger to the fire is
S
2
0
∘
W
\mathrm{S} 20^{\circ} \mathrm{W}
S
2
0
∘
W
. How far is the first ranger from the fire?
\newline
How far is the first ranger from the fire?
\newline
mi (Round to the nearest tenth of a mile.)
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2
2
2
. An airplane takes off
200
200
200
yards in front of a
180
180
180
yard building.
\newline
Find the angle of elevation for the plane so it does not hit the building.
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A man starts walking north at
2
ft
s
2\frac{\text{ft}}{\text{s}}
2
s
ft
from a point
P
P
P
. Five minutes later a woman starts walking south at
5
ft
s
5\frac{\text{ft}}{\text{s}}
5
s
ft
from a point
500
ft
500\text{ft}
500
ft
due east of
P
P
P
. At what rate are the people moving apart
15
min
15\text{min}
15
min
after the woman starts walking? (Round your answer to two decimal places.)
\newline
ft
/
s
\text{ft}/\text{s}
ft
/
s
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Find the volume of a cube with a side length of
4
m
4 \mathrm{~m}
4
m
, to the nearest tenth of a cubic meter (if necessary).
\newline
Answer:
m
3
\mathrm{m}^{3}
m
3
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The hypotenuse of a right triangle measures
17
c
m
17 \mathrm{~cm}
17
cm
and one of its legs measures
15
c
m
15 \mathrm{~cm}
15
cm
. Find the measure of the other leg. If necessary, round to the nearest tenth.
\newline
Answer:
□
\square
□
c
m
\mathrm{cm}
cm
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One of the legs of a right triangle measures
7
c
m
7 \mathrm{~cm}
7
cm
and its hypotenuse measures
9
c
m
9 \mathrm{~cm}
9
cm
.
\newline
Find the measure of the other leg. If necessary, round to the nearest tenth.
\newline
Answer:
□
\square
□
c
m
\mathrm{cm}
cm
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One of the legs of a right triangle measures
9
c
m
9 \mathrm{~cm}
9
cm
and the other leg measures
12
c
m
12 \mathrm{~cm}
12
cm
.
\newline
Find the measure of the hypotenuse. If necessary, round to the nearest tenth.
\newline
Answer:
□
\square
□
c
m
\mathrm{cm}
cm
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The hypotenuse of a right triangle measures
10
c
m
10 \mathrm{~cm}
10
cm
and one of its legs measures
9
c
m
9 \mathrm{~cm}
9
cm
. Find the measure of the other leg. If necessary, round to the nearest tenth.
\newline
Answer:
□
\square
□
c
m
\mathrm{cm}
cm
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One of the legs of a right triangle measures
12
c
m
12 \mathrm{~cm}
12
cm
and its hypotenuse measures
13
c
m
13 \mathrm{~cm}
13
cm
. Find the measure of the other leg. If necessary, round to the nearest tenth.
\newline
Answer:
□
\square
□
c
m
\mathrm{cm}
cm
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One of the legs of a right triangle measures
4
c
m
4 \mathrm{~cm}
4
cm
and its hypotenuse measures
5
c
m
5 \mathrm{~cm}
5
cm
.
\newline
Find the measure of the other leg. If necessary, round to the nearest tenth.
\newline
Answer:
□
\square
□
c
m
\mathrm{cm}
cm
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The hypotenuse of a right triangle measures
13
c
m
13 \mathrm{~cm}
13
cm
and one of its legs measures
5
c
m
5 \mathrm{~cm}
5
cm
. Find the measure of the other leg. If necessary, round to the nearest tenth.
\newline
Answer:
□
\square
□
c
m
\mathrm{cm}
cm
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One of the legs of a right triangle measures
9
c
m
9 \mathrm{~cm}
9
cm
and the other leg measures
10
c
m
10 \mathrm{~cm}
10
cm
. Find the measure of the hypotenuse. If necessary, round to the nearest tenth.
\newline
Answer:
□
\square
□
c
m
\mathrm{cm}
cm
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One of the legs of a right triangle measures
6
c
m
6 \mathrm{~cm}
6
cm
and its hypotenuse measures
10
c
m
10 \mathrm{~cm}
10
cm
. Find the measure of the other leg. If necessary, round to the nearest tenth.
\newline
Answer:
□
\square
□
c
m
\mathrm{cm}
cm
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The hypotenuse of a right triangle measures
15
c
m
15 \mathrm{~cm}
15
cm
and one of its legs measures
12
12
12
c
m
\mathrm{cm}
cm
. Find the measure of the other leg. If necessary, round to the nearest tenth.
\newline
Answer:
□
\square
□
c
m
\mathrm{cm}
cm
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The hypotenuse of a right triangle measures
10
c
m
10 \mathrm{~cm}
10
cm
and one of its legs measures
6
c
m
6 \mathrm{~cm}
6
cm
. Find the measure of the other leg. If necessary, round to the nearest tenth.
\newline
Answer:
□
\square
□
c
m
\mathrm{cm}
cm
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The hypotenuse of a right triangle measures
9
c
m
9 \mathrm{~cm}
9
cm
and one of its legs measures
2
c
m
2 \mathrm{~cm}
2
cm
. Find the measure of the other leg. If necessary, round to the nearest tenth.
\newline
Answer:
□
\square
□
c
m
\mathrm{cm}
cm
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Whenever she visits Kensington, Elise has to drive
8
8
8
miles due north from home, Whenever she visits Richmond, she has to drive
5
5
5
miles due east from home, How far apart are Kensington and Richmond, measured in a straight line? If necessary, round to the nearest tenth.
\newline
miles
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Three ballet dancers are positioned on stage. Kate is straight behind Maura and directly left of Craig. If Maura and Kate are
7
7
7
meters apart, and Craig and Maura are
9
9
9
meters apart, what is the distance between Kate and Craig? If necessary, round to the nearest tenth.
\newline
meters
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Layla leans a
26
26
26
-foot ladder against a wall so that it forms an angle of
6
1
∘
61^{\circ}
6
1
∘
with the ground. How high up the wall does the ladder reach? Round your answer to the nearest tenth of a foot if necessary.
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Two boxes have the same volume. One box has a base that is
5
c
m
5 \mathrm{~cm}
5
cm
by
5
c
m
5 \mathrm{~cm}
5
cm
. The other box has a base that is
10
c
m
10 \mathrm{~cm}
10
cm
by
10
c
m
10 \mathrm{~cm}
10
cm
.
\newline
How many times as tall is the box with the smaller base?
\newline
□
\square
□
times as tall
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Yoku is putting on sunscreen. He uses
2
m
l
2 \mathrm{ml}
2
ml
to cover
50
c
m
2
50 \mathrm{~cm}^{2}
50
cm
2
of his skin. He wants to know how many milliliters of sunscreen
(
c
)
(c)
(
c
)
he needs to cover
325
c
m
2
325 \mathrm{~cm}^{2}
325
cm
2
of his skin.
\newline
How many milliliters of sunscreen does Yoku need to cover
325
c
m
2
325 \mathrm{~cm}^{2}
325
cm
2
of his skin?
\newline
m
l
\mathrm{ml}
ml
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Eliana drove her car
81
k
m
81 \mathrm{~km}
81
km
and used
9
9
9
liters of fuel. She wants to know how many kilometers
(
x
)
(x)
(
x
)
she can drive on
22
22
22
liters of fuel. She assumes her car will continue consuming fuel at the same rate.
\newline
How far can Eliana drive on
22
22
22
liters of fuel?
\newline
k
m
\mathrm{km}
km
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Ted likes to run long distances. He can run
20
k
m
20 \mathrm{~km}
20
km
in
95
95
95
minutes. He wants to know how many kilometers
(
k
)
(k)
(
k
)
he will go if he runs at the same pace for
285
\mathbf{2 8 5}
285
minutes.
\newline
How far will Ted run in
285
285
285
minutes?
\newline
k
m
\mathrm{km}
km
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Whitney is making a strawberry cake for her brother's birthday. The cake pan is a right rectangular prism
20
c
m
20 \mathrm{~cm}
20
cm
wide by
28
c
m
28 \mathrm{~cm}
28
cm
long. Whitney puts
1848
c
m
3
1848 \mathrm{~cm}^{3}
1848
cm
3
of batter into the pan.
\newline
How deep is the cake batter?
\newline
cm
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Rosa is building a guitar. The second fret is
33.641
m
m
33.641 \mathrm{~mm}
33.641
mm
from the first fret. The third fret is
31.749
m
m
31.749 \mathrm{~mm}
31.749
mm
from the second fret.
\newline
How far is the third fret from the first fret?
\newline
m
m
\mathrm{mm}
mm
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Salma went on a walk in her neighborhood.
\newline
First, she walked on a straight road for
2
k
m
2 \mathrm{~km}
2
km
. The direction of the road is a
2
0
∘
20^{\circ}
2
0
∘
rotation from east.
\newline
Then, she turned into a different road whose direction is a
10
0
∘
100^{\circ}
10
0
∘
rotation from east. She walked on that road for
3
k
m
3 \mathrm{~km}
3
km
.
\newline
How far is Salma from her starting point at the end of the walk? Round your answer to the nearest tenth. You can round intermediate values to the nearest hundredth.
\newline
k
m
\mathrm{km}
km
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Cecelia went on a hiking trip. The first day she walked
27
27
27
kilometers. Each day since, she walked
2
3
\frac{2}{3}
3
2
of what she walked the day before.
\newline
What is the total distance Cecelia has traveled by the end of the
5
th
5^{\text {th }}
5
th
day?
\newline
Round your final answer to the nearest kilometer.
\newline
k
m
\mathrm{km}
km
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Antonio's toy boat is bobbing in the water under a dock. The vertical distance
H
H
H
(in
c
m
\mathrm{cm}
cm
) between the dock and the top of the boat's mast
t
t
t
seconds after its first peak is modeled by the following function. Here,
t
t
t
is entered in radians.
\newline
H
(
t
)
=
5
cos
(
2
π
3
t
)
H(t)=5 \cos \left(\frac{2 \pi}{3} t\right)
H
(
t
)
=
5
cos
(
3
2
π
t
)
\newline
How long does it take the toy boat to bob down from its peak to a height of
−
35
c
m
-35 \mathrm{~cm}
−
35
cm
?
\newline
Round your final answer to the nearest tenth of a second.
\newline
seconds
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Angela is building a pool that is
25
\mathbf{2 5}
25
meters long. She has selected tiles to put around the edge of the pool. If each tile is
8
8
8
inches long, how many tiles does she need for one
25
25
25
-meter side of the pool?
\newline
(
1
1
1
meter
≈
3.28
\approx 3.28
≈
3.28
feet)
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The great snipe is the fastest known migratory bird. Scientists found that the bird can fly non-stop for approximately
4200
4200
4200
miles (mi). If the bird's speed is an average of
98
98
98
kilometers per hour
(
k
m
h
r
)
\left(\frac{\mathrm{km}}{\mathrm{hr}}\right)
(
hr
km
)
, how many hours can the great snipe fly without stopping?
\newline
(Round the answer to the nearest tenth.
1
1
1
kilometer
≈
0.62
\approx 0.62
≈
0.62
miles)
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A rectangular prism and cube have equal volumes. The length of the rectangular prism is
12
12
12
centimeters
(
c
m
)
(\mathrm{cm})
(
cm
)
and its width is
8
c
m
8 \mathrm{~cm}
8
cm
. If each side of the cube is
12
c
m
12 \mathrm{~cm}
12
cm
, then what is the height of the rectangular prism in centimeters?
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