Pre-TestUnit8:GeometryApplicationsKEY
You may use a calculator.
Answer the following questions. (Partial credit at teacher discretion)
1. What is the IF-THEN statement for the Pythagorean Theorem? (2 pts)
IF a triangle is a right triangle, then a + b = c where a and b are leg lengths and c is hypotenuse length.
2. What is the Pythagorean Theorem used for? (3 pts)
We can use the Pythagorean Theorem to find a missing side length in a right triangle.
3. What is the IF-THEN statement for the Pythagorean Theorem Converse? (2 pts)
IF a + b = c in a triangle, then it is a right triangle.
4. What is the Pythagorean Theorem Converse used for? (3 pts)
The Pythagorean Theorem Converse can be used to check for right angles in a triangle.
Determine if each of the following is a right triangle or not using the Pythagorean Theorem. (4 pts; 3 pts for
set-up/work or explanation, 1 pts for correct answer)
5.
6.
26
24
27
21
20
10
Yes, it’s a 5, 12, 13 multiplied by 2.
No, 20 + 21 ≠ 27
Find the length of the missing side of each right triangle. Round your answer to three decimal places if
necessary. (4 pts; 2 pts for set-up/work or explanation, 2 pts for correct answer)
7.
8.
13
5
= 12
23
7
≈ 24.042
Find the value of the variable. Round your answer to three decimal places if necessary. (4 pts; 2 pts for setup/work or explanation, 2 pts for correct answer)
9. The following cone has a radius of 6 and a
height of 8. What is , the slant height?
10. The following pyramid has a square base that is
50 on each side. The slant height is 50. What
is , the height of the pyramid?
slant
height
slant
height
"
10
43.301
Determine the distance between the given points. Round your answer to three decimal places if necessary.
(4 pts; 2 pts for set-up/work or explanation, 2 pts for correct answer)
11. 6, 3 and 6,2
13
12. 6,4 and 8, 4
16.125
Solve the following problems. (4 pts; 2 pts for set-up/work, 2 pts for correct final answer)
13. A hospital helicopter must go pick up a patient that is six miles west and eight miles north of the hospital.
How many miles total will the helicopter travel to pick up the patient and bring him back to the hospital?
20 miles
14. A nature area has a rectangle field that is 10 miles by 5 miles and wants to put a fence along the diagonal of
the field that will costs $1,000 per mile. How much will the fence cost to the nearest dollar?
$11,180
Find the given dimension of each shape either using # $. %& or giving your answer in terms of #. Round your
answer to two decimal places if necessary. (5 pts; 3 pts for set-up/work, 2 pts for correct answer)
15. Find the volume
10
25
16. Find the volume
17. Find the volume
3
12
5
2500'( 7850(
100'( 314(
36'( 113.04(
18. Find the radius of a cylindrical fire hose that is 200 long and has a volume of 39.25 (.
0.25
19. Find the height of a waffle cone for ice cream that has a volume of 25.12( and a radius of 2.
6
20. Find the radius of a spherical water balloon with a volume of 904.32( .
6
3
Find the volume of each shape either using # $. %& or giving your answer in terms of #. Round your answer
to two decimal places if necessary. (10 pts; 4 pts for work/volume of each shape, 2 pts for final answer)
21. A cylindrical propane gas tank with half spheres on either end that is 9 long (not including the half spheres)
and has a 3 radius
81' + 36' = 117' ( 367.38 (
OR
*+,-" 254.34 ( +./-"- 113.04 (
9
22. A caulking gun with a radius of 1, cone height of 6, and cylinder height of 20
6
20
20' + 2' = 22'( 69.08(
OR
*+,-" 62.8(
+*0- 6.28(
4
Lesson 8.1
Unit 8 Homework Key
1. What is the Pythagorean Theorem in your own words?
If a triangle is a right triangle, then a + b c where a and b are the side lengths of the legs and c is the length
of the hypotenuse.
2. What does the Pythagorean Theorem allow us to do?
It allows us to find missing side lengths of a right triangle.
3. What is the Pythagorean Theorem Converse in your own words?
If a + b c in a triangle, then it is a right triangle.
4. What does the Pythagorean Theorem Converse allow us to do?
It allows us to determine if an angle is a right angle or not.
5. The door to your bathroom has never closed well. In fact, every time you try to use the bathroom, the cats
bust open the door because it simply won’t latch. You look at the door and it appears that the door frame is
slightly tilted. The person who built your house claims that can’t be true because he measured your door frame
and found it be an exact right angle. He claims what you’re seeing is an optical illusion.
32
a. Without having a protractor, what could you do to see if he is correct
without having a protractor?
Use the Pythag Converse to see if it works.
b. If you knew the door frame measurements were as pictured to the right,
did the builder install your door frame correctly at a right angle?
No because 32 + 81 86 .
81
86
6. Bob is building a triangular garden and needs fencing around it to keep the rabbits out. He has one section of
fence measuring 40 ft, another measuring 42 ft, and a third measuring 58 ft. Bob says that after the fence is
complete it will make a right triangle using the following argument: “First, I’ll set-up the longest section of fence.
Next, I’ll attach the other two sections to either end of the long one. Finally, I’ll swing the two shorter sections
together. Since they must meet together, that makes it a right triangle.”
a. Is Bob correct that the garden fence will make a right triangle?
Yes, it will make a right triangle since 40 + 42 58.
b. If so, is Bob’s argument correct for why it will make a right triangle?
No, simply meeting will just make a triangle, not necessarily a right triangle.
c. What would be a better argument?
Use the Pythagorean Theorem Converse.
5
Determine if the following triangles are right triangles or not using the Pythagorean Theorem.
7.
8.
9.
17
8
25
15
24
6
8
Yes, right triangle
10.
7
7
Yes, right triangle
11.
No, not right triangle
12.
14
10
41
10
40
8
No, not right triangle
9
Yes, right triangle
6
8
Yes, right triangle
13. 1 12
14. 1 122
15. 1 10
16
352
24
25
372
27
No, not right triangle
Yes, right triangle
No, not right triangle
16. 1 20
17. 1 52
18. 1 5
21
122
12
29
172
13
Yes, right triangle
No, not right triangle
6
Yes, right triangle
Lesson 8.2
Find the length of the missing side of each right triangle. Round your answers to three decimal places if
necessary.
1.
2.
40
1
3.
29
20
4
1
3
9
1 38.974
1 21
4.
5.
25
5
6.
73
55
5
10
3
22.913
48
5.831
7.
8.
9.
3
152
+
28
40
30
45
82
4
+ 172
3 53
4 26.458
10.
11.
12.
26
5
13
12
61
4
60
10
5 24
5
4 11
7
13.
14.
15.
45
202
"
4
28
40
72
18.7352
21
" 20.616
4 35
Solve the following problems. Round your answers to the nearest whole number when necessary.
16. You’re locked out of your house, and the only open window is on the second floor 25 feet above the ground.
There are bushes along the side of the house that force you to put the base of the ladder 7 feet away from the
base of the house. How long of a ladder will you need to reach the window?
≈ 26--
17. Shae takes off from her house and runs 3 miles north and 4 miles west. Tired, she wants to take the shortest
route back. How much farther will she have to run if she heads straight back to her house?
5-6
18. Televisions are advertised by the length of their diagonals. If a 42 inch television measures 18 inches high,
how wide is the television?
≈ 38-6
19. A soccer field is 100 yards by 60 yards. How long is the diagonal of the field?
≈ 117+1",6
20. Leonard walks 14 meters south and 48 meters east to get to school. If he takes the straight path home after
school, how far will he have to walk?
50--"6
21. You place a 24 foot ladder 10 feet away from the house. The top of the ladder just reaches a window on the
second floor. How high off the ground is the window?
≈ 22--
22. The dimensions of a basketball court are 74 feet and 42 feet. What is the length of the diagonal of the court?
≈ 85--
8
23. Televisions are advertised by the length of their diagonals. If a TV measures 22 inches high and 45 inches
wide, by what size will the TV be advertised.
≈ 50
24. A rectangular garden measures 5 feet wide by 12 feet long. If a hose costs $5 per foot, how much would it
cost to place a hose through the diagonal of the garden?
$65
25. A football field is 160 feet wide and 360 feet long. The coach wants to put spray paint along the diagonal of
the field. If the spray paint costs approximately $1 per foot of coverage, how much should the coach budget for
spray paint?
≈ $394
26. A rectangular park measures 8 miles long by 6 miles wide. The park director wants to put a fence along both
sides of the trail that runs diagonally through the park. If the fence costs $150 per mile, how much will it cost to
buy the fence?
$3000
27. A rectangular pool has a diagonal of 17 yards and a length of 15 yards. If the paint costs $2 per yard of
coverage, how much will it cost the owner to paint the width of both ends of the pool?
$32
28. A rectangular dog pen is 3 meters by 4 meters. If a chain costs $1.75 per meter, how much would it cost to
put a chain along the diagonal of the pen?
$8.75
29. Architects built a doorway that was 4 feet wide by 7 feet tall. The diagonal measured 7.3 feet. Are the angles
in the doorway right angles?
No
30. A rectangular garden measures 3 meters wide by 4 meters long. The diagonal of the garden measures 5
meters. Are the angles in the garden right angles?
Yes
9
Lesson 8.3
Use the picture below to find information about the pyramid with a square base in problems 1-14. Round your
answers to three decimal places if necessary.
slant
height
1
1. The pyramid has a square base that is 70 on each side. The slant height is
37. What is , the height of the pyramid?
12
2. The pyramid has a square base that is 120 on each side. The slant height is
61. What is , the height of the pyramid?
11
3. The pyramid has a square base that is 50 on each side. The slant height is
30. What is , the height of the pyramid?
16.583
4. The pyramid has a square base that is 14 on each side. The slant height is 25. What is , the height?
24
5. The pyramid has a square base that is 14 on each side. The height is 24. What is , the slant height?
25
6. The pyramid has a square base that is 24 on each side. The height is 5. What is , the slant height?
13
7. The pyramid has a square base that is 70 on each side. The height is 10. What is , the slant height?
36.401
8. The pyramid has a square base that is 26 on each side. The height is 82. What is , the slant height?
≈ 83.024
9. The height of the pyramid is 15, and the slant height is 39. Find the value of 1 in the diagram.
36
10. The height of the pyramid is 80, and the slant height is 82. Find the value of 1 in the diagram.
18
11. The slant height is 17 and the height is 8. What is 6, the side length of the base?
30
12. The slant height is 10 and the height is 8. What is 6, the side length of the base?
12
13. The slant height is 26 and the height is 10. What is 6, the side length of the base?
48
14. The slant height is 50 and the height is 32. What is 6, the side length of the base?
76.837
10
Use the picture below to find information about the pyramid in problems 15-26. Round your answers to three
decimal places if necessary.
slant
height
15. The cone has a radius of 12 and a height of 5. What is , the slant
height of the cone?
13
"
16. The cone has a radius of 15 and a height of 8. What is , the slant
height of the cone?
17
17. The cone has a radius of 24 and a height of 70. What is , the slant
height of the cone?
74
18. The cone has a radius of 40 and a height of 42. What is , the slant height of the cone?
58
19. The cone has a radius of 30 and a slant height of 34. What is , the height of the cone?
16
20. The cone has a radius of 33 and a slant height of 65. What is , the height of the cone?
56
21. The cone has a radius of 16 and a slant height of 20. What is ℎ, the height of the cone?
12
22. The cone has a radius of 30 and a slant height of 50. What is , the height of the cone?
40
23. The cone has a height of 16 and a slant height of 65. What is ", the radius of the cone?
63
24. The cone has a height of 48 and a slant height of 50. What is ", the radius of the cone?
14
25. The cone has a height of 4 and a slant height of 6. What is ", the radius of the cone?
≈ 4.472
26. The cone has a height of 14 and a slant height of 55. What is ", the radius of the cone?
53.188
11
Use the picture below to find lengths of segments in the rectangular prism in problems 27-38. Round your
answers to three decimal places if necessary.
D
9999 is 6 and the length of 9999
9999 .
27. The length of 78
8* is 8. Find the length of 7*
10
9999 is 40 and the length of 9999
9999 .
28. The length of 78
8* is 42. Find the length of 7*
58
C
A
B
9999 is 23 and the length of 9999
9999 .
29. The length of 78
8* is 70. Find the length of 7*
≈ 73.682
9999 is 7 and the length of 9999
9999 .
30. The length of 78
8* is 7. Find the length of 7*
9.899
9999 is 13 and the length of 9999
9999.
31. The length of 7*
:* is 84. Find the length of 7:
85
9999.
9999 is 5 and the length of 9999
:* is 12. Find the length of 7:
32. The length of 7*
13
9999.
9999 is 11 and the length of 9999
33. The length of 7*
:* is 30. Find the length of 7:
31.953
9999.
9999 is 5 and the length of 9999
34. The length of 7*
:* is 4. Find the length of 7:
6. 403
9999.
9999 is 4, the length of 9999
35. The length of 78
8* is 3 and the length of 9999
:* is 12. Find the length of 7:
13
9999.
9999 is 12, the length of 9999
36. The length of 78
8* is 5 and the length of 9999
:* is 84. Find the length of 7:
85
9999.
9999 is 2, the length of 9999
37. The length of 78
8* is 3 and the length of 9999
:* is 10. Find the length of 7:
10.630
9999 is 6 the length of 9999
9999.
38. The length of 78
8* is 8 and the length of 9999
:* is 50. Find the length of 7:
50.990
12
Lesson 8.4
Determine the distance between the given points. Round your answers to three decimal places if necessary.
1. 1, 3 and 4, 7
2.
5;6
2.
3, 3 and 2, 9
13;6
3. 2, 5 and 3, 8
5.831;6
4.
3, 3 and 3, 3
8.485;6
5. 3, 2 and 5, 0
2.828;6
6.
3, 9 and 3, 9
18;6
13
7. 2, 1) and (3, −3)
≈ 4.123;6
8.
9. 1, 1) and (7, 9)
10;6
10. 8, 2) and (6, 2)
14;6
11. 4, 6) and (6, 2)
≈ 10.770;6
12. 2, 4 and 5, −2)
≈ 6.708;6
13. 5, −3) and (6, 6)
≈ 14.213;6
14. 5, 4) and (7, 3)
≈ 12.042;6
15. 9, −3) and (−4, 4) ≈ 8.602;6
16. 2, −4 and 5, 4)
≈ 8.544;6
17. 0, 7) and (4, 2)
18. 8, 7) and (7, −5)
≈ 19.209;6
≈ 6.403;6
14
4, −2) and (7, 2)
5;6
Lesson 8.5
Answer the following questions either using # $. %& or giving your answer in terms of #. Round your answer
to the nearest hundredth where necessary.
1. Find the volume of a cylinder with a radius of 3 and a height of 10.
90'( ≈ 282.6(
2. Find the volume of a cylinder with a radius of 10 and a height of 2.
200'( ≈ 628(
"
3. Find the volume of a cylinder with a radius of 5 and a height of 15.
375'( 1177.5(
4. Find the volume of a cylinder with a diameter of 22 and a height of 5.
605'( 1899.7(
5. Find the volume of a cylinder with a diameter of 4 and a height of 1.
4' ( ≈ 12.56 (
6. Find the volume of a cylinder with a radius of 9 and a height of 9.
729'( ≈ 2289.06(
7. Find the volume of a can of green beans with a radius of 3 and a height of 8.
72'( ≈ 226.08(
8. Find the volume of a cylindrical can of oatmeal with a radius of 8 and a height of 45.
2880'( ≈ 9043.2(
9. Find the volume of a cylindrical water bottle with a diameter of 4 and a height of 30.
120'( 376.8(
10. Find the volume of a can of Pepsi with a diameter of 2 and a height of 3.5.
3.5'( 10.99(
11. Find the volume of a water pipe with a radius of 0.75 and a length of 16.
9' ( 28.26 (
12. Find the volume of a straw used for drinking with a radius of 2 and a height of 170.
680'( ≈ 2135.2(
15
18.
19.
20.
27.
28.
29.
30.
31.
13. Find the volume of a cone with a radius of 3 and a height of 10 .
30' ( ≈ 94.2 (
14. Find the volume of a cone with a radius of 10 and a height of 3 .
100' ( 314 (
15. Find the volume of a cone with a radius of 5 and a height of 15 .
125' ( 392.5 (
ℎ
16. Find the volume of a cone with a radius of 12 and a height of 5 .
240' ( 753.6 (
"
17. Find the volume of a cone with a diameter of 4 and a height of 9 .
12' ( 37.68 (
Find the volume of a cone with a diameter of 18 and a height of 9 .
243' ( 763.02 (
Find the volume of a waffle cone for ice cream with a radius of 4 and a height of 12 .
64' ( 200.96 (
Find the volume of a cone birthday hat with a radius of 2 and a height of 9 .
12' ( 37.68 (
21. Find the volume of a funnel with a diameter of 10 and a height of 9 .
75' ( 235.5 (
22. Find the volume of a sphere with a diameter of 6 .
36' ( 113.04 (
23. Find the volume of a sphere with a diameter of 18 .
972' ( 3052.08 (
"
24. Find the volume of a sphere with a radius of 6 .
288' ( 904.32 (
25. Find the volume of a sphere with a radius of 12 .
2304' ( 7234.56 (
26. Find the volume of a sphere with a radius of 2 .
10.67' ( 33.49 (
Find the volume of a sphere with a radius of 5 .
166.67' ( 523.33 (
Find the volume of a mini basketball with a radius of 3.5 .
57.17' ( 179.51 (
Find the volume of the Earth with a diameter of approximately 12, 756 2.
3.46 < 10== ' 2( 1.09 < 10= 2(
Find the volume of the moon with a diameter of approximately 3475 2.
6.99 < 10> ' 2( 2.20 < 10=? 2(
Find the volume of a gumball with a radius of 3 .
36' ( 113.04 (
16
Lesson 8.6
Answer the following questions using # $. %&.
necessary.
Round your answer to the nearest hundredth where
1. Find the height of a cylinder with a volume of 30( and a radius of 1.
9.55
2. Find the height of a cylinder with a volume of 100( and a radius of 2.
7.96
3. Find the height of a cylinder with a volume of 720' ( and a radius of 6.
20
4. Find the height of a cylinder with a volume of 1215'( and a radius of 9.
15
5. Find the radius of a cylinder with a volume of 950( and a height of 10.
" 5.5
6. Find the radius of a cylinder with a volume of 208( and a height of 4.
" 4.07
7. Find the radius of a cylinder with a volume of 108' ( and a height of 12.
" 3
8. Find the radius of a cylinder with a volume 686( and a height of 14.
" 3.95
9. Find the height of a cone with a volume of 150( and a radius of 10.
1.43
(
10. Find the height of a cone with a volume of 21 and a radius of 4.
1.25
11. Find the radius of a cone with a volume of 175( and a height of 21.
" ≈ 2.82
12. Find the radius of a cone with a volume of 196'( and a height of 12.
" 7
13. Find the radius of a sphere with volume ≈ 113.04( .
" 3
14. Find the radius of a sphere with volume 904.32( .
" 6
15. Find the radius of a sphere with volume 3052.08( .
" 9
16. Find the radius of a sphere with volume ≈ 4.1869 ( .
" 1
17
Lesson 8.7
Answer the following questions using # $. %& and rounding your answer to the nearest hundredth where
necessary.
1. Find the volume of a cone used for the tip of a rocket with a diameter of 12+,6 and a height of
15+,6.
180'+1",6 ( 565.2+1",6 (
2. Find the volume of a pencil with a radius of 0.5, a cone height of 3, and a cylinder height
of 14.
3.75'( ≈ 11.78(
3. Find the volume of a model rocket with a radius of 1, a cone height of 3, and a cylinder
height of 8.
@#AB$ ≈ CD. CEAB$
Visual for
questions
1-10
4. Find the volume of a caulking gun with a radius of 2, a cone height of 3, and a cylinder
height of 20.
84'( 263.76(
5. Find the volume of a crayon with a radius of 2, a cone height of 21, and a cylinder height
of 80.
348'( 1092.72(
6. Find the volume of a model jet with a radius of 1, a cone height of 3, and a cylinder height
of 6.
7' ( ≈ 21.98 (
7. Find the radius of a pencil with a volume of 110'( , a cone height of 3, and a cylinder height of
10.
" 3.16
8. Find the radius of a model rocket with a volume of 2500( , a cone height of 6, and a cylinder height of
25.
" 5.43
9. Find the cylinder height of a caulking gun with a volume of 300( , a cone height of 6, and a radius of
2.
h 21.89
10. Find the cone height of a pencil with a volume of 750(, a radius of 3, and a cylinder height of
25.
h 4.62
18
H
11. Find the volume of a propane gas tank with half spheres on either end
that has a radius of 3 and a length () of 7.
Visual for
questions
11-12
99' ( 310.86 (
12. Find the volume of a submarine with half spheres on either end that has a
radius of 6 and a length () of 15.
828'( ≈ 2599.92(
13. Find the volume of a grain silo with a half sphere on one end that has a
diameter of 6 and a height (ℎ) of 15.
153'( 480.42(
Visual for
questions
13-14
G
14. Find the volume of a grain silo with a half sphere on one end that has a
diameter of 6 and a height (ℎ) of 35.
333' ( 1045.62 (
19
ReviewUnit8:GeometryApplicationsKEY
You may use a calculator.
Unit 8 Goals
• Explain a proof of the Pythagorean Theorem and its converse. (8.G.6)
• Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and
mathematical problems in two and three dimensions. (8.G.7)
• Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. (8.G.8)
• Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and
mathematical problems. (8.G.9)
Determine if the following triangles are right triangles or not using the Pythagorean Theorem.
1.
2.
20
29
7
8
6
No, 6 7 8
21
Yes, 21 + 20 29
Find the length of the missing side of each right triangle. Round to three decimal places if necessary.
3.
4.
12
4
45
9
4 15
5.
50
5
2
2 21.794
26
10
5 24
Determine the distance between the given points. Round to three decimal places if necessary.
6. 0, 8 and 6, 0
10;6
7. 1, 5 and 6, 5
20
11.180;6
Find the value of the variable.
8. The following cone has a radius of 11 and a
slant height of 61. What is , the height?
9. The following cone has a height of 20 and a
slant height of 29. What is ", the radius?
slant
height
slant
height
ℎ
"
"
60
" 21
10. The following pyramid has a square base that is
30 on each side. The height is 8. What is ,
the slant height of the pyramid?
11. The following pyramid has a square base. The
height is 12 and the slant height is 20. What is
6, the side length of the base of the pyramid?
slant
height
slant
height
ℎ
ℎ
17
6 32
Solve the following problems.
12. Firefighters position an 85-foot ladder 13 feet away from the building. The top of the ladder just reaches a
window on the fourth floor. How high off the ground is the window?
84
13. The school is located 9 meters north and 40 meters west of Kiley’s house. Kiley walks through her neighbors’
yards, so she can take the shortest route possible (a straight line). How far does she have to travel if she walks to
and from school?
82--"6
14. An open field is 85 meters wide and 105 meters long. The owner wants to put spray paint along both
diagonals of the field. If the spray paint costs approximately $2 per meter of coverage, how much should the
owner budget for spray paint?
≈ $540.36
21
Find the volume of the given shapes using # $. %& for your answers or in terms of #.
15.
16.
17.
4
9
6
8
2
I 128' ≈ 401.92(
I 288' ≈ 904.32(
I 12' ≈ 37.68 (
18. Cylinder
12
" 2
19. Cone
15
" 3
20. Sphere
" 3
I 48' ≈ 150.72(
I 45' ≈ 141.3(
I 36' ≈ 113.04(
Find the missing dimension of the given shapes using # $. %& for your answers or in terms of #.
21. I ≈ 401.92(
22. I 3052.08(
23. I 37.68 (
"
ℎ
"
8
2
" 4
" 9
9
24. Cylinder
12
" ?
I 150.72(
25. Cone
?
" 3
I 141.3(
26. Sphere
" ?
I 113.04(
" 2
15
" 3
22
Find the volume of the given shapes in terms of # or using # $. %&.
27. A spherical volleyball with a radius of 15.
I 4500'( 14130(
28. A cylindrical water bottle with a height of 10 and a radius of 1.
I 10'( 31.4(
29. An ice cream cone with a height of 6 and radius of 2.
I 8'( ≈ 25.12(
9
30. A grain silo as pictured:
I 900' + 108'
= 1008' ≈ 3165.12 (
6
25
31. A propane tank as pictured:
3
I 90' + 36'
= 126' ≈ 395.64(
10
23