Chapter 8 Practice Test #1 - M2 Mathdavidbeckeyeipbcc.weebly.com/.../chapter_8_practice_test.pdf ·...

Chapter 8 Practice Test #1

Class Name : M29 Geometry - 2019-20 Term 2 Instructor Name : Mr. Beckey

Student Name : _____________________ Instructor Note :

1. In the figure below, there are three right trangles. Complete the following.

2. In the figure below, find the exact value of . (Do not approximate your answer.)

3. In the figure below,

P

QR

S

(a) Write a similiarity statement relating the three right triangles.

~P R Q ~

(b) Complete each proportion.

=R Q

P Q R Q=

P R

P S

P Q

y

ADC CDB ACB

y

7

2

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3. In the figure below, .

Fill in the following blanks using the lengths , , , , and .Part 1: Complete the proportions.

Part 2: Use the method of cross products to rewrite the equations in Part 1.

__ __Part 3: Use Part 2 to fill in the blanks.

__ __Part 4: Factor the right-hand side of Part 3.

__ __Part 5: Use the Segment Addition Property.

__Part 6: Use Part 5 to rewrite the equation in Part 4.

__Part 7: Simplify.

__

4. The right triangle below has legs of length and

ADC ~ CDB ~ ACB

a b c x y

=c

b

b

=c

a

a

=b2 ·c =a2 ·c

=+a2 b2 ·c ·+c

=+a2 b2 c +

=+x y

=+a2 b2 c

=+a2 b2 2

=a 16 =b 5

B A

C

D

ba

c

x y

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4. The right triangle below has legs of length and .

The hypotenuse has length .

Four copies of this triangle are arranged as follows.

The hypotenuses form a square of side length and area .

Answer the questions below to find how , , and are related.

Part 1: Compute the total combined area of the four triangles: __Part 2: Compute the area of the large (outer) square: __Part 3: Using your answers in Parts 1 and 2, find the area of the small (inner) square.

__

Part 4: We are given the side lengths and . Compute .

__

Part 5: Use , , or to complete the statement below.

__

5. For the following right triangle, find the side length

=a 16 =b 5c

c c2

a b c

=c2

=a 16 =b 5 +a2 b2

=+a2 b2

< > =

+a2 b2 c2

x

=b 5

=a 16c

c c

cc

=b 5

=b 5

=b 5

=b 5

=a 16

=a 16

=a 16

=a 16

=Area c2

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5. For the following right triangle, find the side length .

6. For the following right triangle, find the side length . Round your answer to the nearest hundredth.

7. A - ladder leans against the side of a house. The bottom of the ladder is from the side of the house.How high is the top of the ladder from the ground? If necessary, round your answer to the nearest tenth.

8. Below are two triangles with their side lengths shown.

x

x

12 ft 6 ft

12

35x

x 8

9

12

6

?

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8. Below are two triangles with their side lengths shown.

Answer the questions about each triangle.

Compute the sum of the squares of theshorter lengths.

______

Compute the square of the longestlength.

______

What kind of triangle is it?

Acute triangle

Right triangle

Obtuse triangle

Compute the sum of the squares of theshorter lengths.

______

Compute the square of the longestlength.

______

What kind of triangle is it?

Acute triangle

Right triangle

Obtuse triangle

9. Find the area of the right triangle.Be sure to include the correct unit in your answer.

10. Determine whether a triangle with the given side lengths is a right triangle.

+82 112 =

122 =

+122 162 =

202 =

8

11 12

16

2012

41 m9 m

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10. Determine whether a triangle with the given side lengths is a right triangle.

Side lengths Right triangle Not a right triangleNot enoughinformation

, ,

, ,

, ,

, ,

11. For the right triangles below, find the exact values of the side lengths and .

If necessary, write your responses in simplified radical form.

12. For the right triangles below, find the values of the side lengths and

14 48 50

10 26 28

9 12 15

10 12 16

h b

c d

60°

30°

h

2

45°

45°

3b

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12. For the right triangles below, find the values of the side lengths and .

Round your answers to the nearest tenth.

13. A right triangle has side lengths , , and as shown below.

Use these lengths to find , , and .

14. Use a calculator to evaluate each expression.Round your answers to the nearest hundredth.

_______

_______

_______

15. Solve for in the triangle. Round your answer to the nearest tenth.

c d

7 24 25tanA sinA cosA

tan12° =

cos77° =

sin49° =

x

60°

30°

2

c

45°

45°

d4

B

AC

7

24

25

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15. Solve for in the triangle. Round your answer to the nearest tenth.

16. Solve the right triangle.

Round your answers to the nearest tenth.

17. Find . Round your answer to the nearest tenth of a degree.

18. Find the area of the given triangle.

x

x

31°

x

8

46°

A

c

b

28

x

1519

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18. Find the area of the given triangle.Round your answer to the nearest tenth. Do not round any intermediate computations.

19. Solve for .

Simplify your answer as much as possible.

20. Solve for .

Simplify your answer as much as possible.

21. The figure below is a right triangle with side lengths and

y

=−2y

8

v

=98v

7

x y z

19

54°

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21. The figure below is a right triangle with side lengths , , and .

Suppose that does not equal .

Complete the following.

Part 1: In , and are

- supplementary.- complementary.- neither complementary nor supplementary.

Part 2: Use , , and to fill in the blanks.Make sure to use the appropriate upper-case or lower-case letters.

Part 3: Select all of the true statements.

None of the above is true.

Part 4: Fill in the blank.

______

22. A kite is flying off the ground, and its string is pulled taut. The angle of elevation of the kite is Find

x y z

m∠X m∠Y

XYZ ∠X ∠Y

x y z

=sinX =sinY

=cosX =cosY

=sinX sinY

=sinX cosY

=cosX cosY

=cosX sinY

=cos 57° sin °

87 ft 58°

X

Y Z

z

x

y

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22. A kite is flying off the ground, and its string is pulled taut. The angle of elevation of the kite is . Findthe length of the string. Round your answer to the nearest tenth.

23. A ramp long rises to a platform that is off the ground. Find , the angle of elevation of the ramp.Round your answer to the nearest tenth of a degree.

24. Looking up, Ann sees two hot air balloons in the sky as shown. She determines that the lower hot air balloon

87 ft 58°

21 ft 16 ft x

58°

?87

x

Platform

1621

Ground

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24. Looking up, Ann sees two hot air balloons in the sky as shown. She determines that the lower hot air balloon

is meters away, at an angle of from the vertical. The higher hot air balloon is meters away, at an

angle of from the vertical. How much higher is the balloon on the right than the balloon on the left?

Do not round any intermediate computations. Round your answer to the nearest tenth.

Note that the figure below is not drawn to scale.

25. Consider a triangle like the one below. Suppose that , , and . (The figure is notdrawn to scale.) Solve the triangle.

Round your answers to the nearest tenth.

26. As shown in the figure below, Jina is standing feet from the base of a leaning tree. The tree is growing at

515 15° 84022°

ABC =A 108° =B 31° =c 3

85

15°22°

515 meters

840 meters

A B

C

c

b a

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26. As shown in the figure below, Jina is standing feet from the base of a leaning tree. The tree is growing at

an angle of with respect to the ground. The angle of elevation from where Jina is standing to the top of the

tree is . Find the length, , of the tree. Round your answer to the nearest tenth of a foot.

27. Consider a triangle like the one below. Suppose that , , and . (The figure is notdrawn to scale.) Solve the triangle.

Carry your intermediate computations to at least four decimal places, and round your answers to the nearesttenth.

28. The side lengths for below are and

8578°

44° x

ABC =c 34 =b 45 =C 30°

ABC a b c

44°

85 ft

78°

x

A

BC

cb

a

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28. The side lengths for below are , , and

The height of is .

Fill in the blanks using the lengths , , , and to derive the Law of Sines.Make sure to use the appropriate upper-case or lower-case letters.

29. Consider a triangle like the one below. Suppose that , , and . (The figure is notdrawn to scale.) Solve the triangle.

Carry your intermediate computations to at least four decimal places, and round your answers to the nearesttenth.

30. Justin is flying two kites. He has feet of string out to one kite and feet out to the other kite. The angle

ABC a b c

ABC h

a b c h

b

B

a

A c

C

h

Part 1: Use trigonometry to fill in the blanks.

=sin B =and sin C

Part 2: Rewrite the equations from Part 1.

=h =⋅ sin B and h ⋅ sin C

Part 3: Use the equations from Part 2 to write an equation

relating and .sinB sinC

=⋅ sin B ⋅ sin C

Part 4: Rewrite the equation from Part 3.

=sin B sin C

ABC =a 62 =b 59 =c 19

95 111

A

BC

cb

a

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30. Justin is flying two kites. He has feet of string out to one kite and feet out to the other kite. The angle

formed by the two strings is as shown in the figure below. Find the distance between the kites.

Carry your intermediate computations to at least four decimal places.Round your answer to the nearest tenth of a foot.

31. The side lengths for are and

95 11131°

ABC a b c

31°95 ft

111 ft

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31. The side lengths for are , , and

The height of the triangle is , and .

Complete the steps below to prove the Law of Cosines.When filling in the blanks, you may use the letters , , , , and

Part 1: Use the Pythagorean Theorem to find .

Part 2: Use the Pythagorean Theorem to find .

Part 3: Use the answer from Part 2 to fill in the blanks.

__ __ __

Part 4: Use the answers from Parts 1 and 3 to fill in the blanks.

__ __ __ __

Part 5: Use trigonometry to fill in the blank.

__

Part 6: Use the answers from Parts 4 and 5 to fill in the blanks.

__ __ __ __

ABC a b c

h =CD x

a b c x h

a2

=a2 +−b x 2 h2 =a2 +−c x 2 h2

=a2 +b2 h2 =a2 +x2 h2

c2

=c2 +−b x 2 h2 =c2 ++x b 2 h2

=c2 +x2 h2 =c2 +−a x 2 h2

=c2 +x2 +h2 −2 ·2 ·

=c2 +2 −2 ·2 ·

=x cosC

=c2 +2 −2 ·2 · cosC

A

a

B

b

C

c

D

h

x

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Chapter 8 Practice Test #1 Answers for class M29 Geometry -2019-20 Term 2

1.

2.

3.

P

QR

S

(a) Write a similiarity statement relating the three right triangles.

~P R Q ~P SR R SQ

(b) Complete each proportion.

=R Q

P Q

SQ

R Q=

P R

P S

P Q

P R

=y 14

© 2019 M cGr aw - H i l l Educat i on. Al l R i ghts R eser ved.C hapter 8 Pr ac t i ce T es t #1 Page 17 / 23

3.

Fill in the following blanks using the lengths , , , , and .Part 1: Complete the proportions.

Part 2: Use the method of cross products to rewrite the equations in Part 1.

Part 3: Use Part 2 to fill in the blanks.

Part 4: Factor the right-hand side of Part 3.

Part 5: Use the Segment Addition Property.

Part 6: Use Part 5 to rewrite the equation in Part 4.

Part 7: Simplify.

4. Part 1: Compute the total combined area of the four triangles:

Part 2: Compute the area of the large (outer) square: Part 3: Using your answers in Parts 1 and 2, find the area of the small (inner) square.

Part 4: We are given the side lengths and . Compute .

Part 5: Use , , or to complete the statement below.

5.

a b c x y

=c

b

b

y=

c

a

a

x

=b2 ·c y =a2 ·c x

=+a2 b2 +·c x ·c y

=+a2 b2 c +x y

=+x y c

=+a2 b2 c c

=+a2 b2 c2

160441

=c2 281

=a 16 =b 5 +a2 b2

=+a2 b2 281

< > =

=+a2 b2 c2

37

B A

C

D

ba

c

x y

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5.

6.

7.

8.

Compute the sum of the squares of the shorterlengths.

Compute the square of the longest length.

What kind of triangle is it?

Acute triangle

Right triangle

Obtuse triangle

Compute the sum of the squares of the shorterlengths.

Compute the square of the longest length.

What kind of triangle is it?

Acute triangle

Right triangle

Obtuse triangle

9. Area:

10.

37

4.12

10.4 ft

+82 112 = 185

122 = 144

+122 162 = 400

202 = 400

180 m2

8

11 12

16

2012

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10.

Side lengths Right triangle Not a right triangleNot enoughinformation

, ,

, ,

, ,

, ,

11.

12.

13.

14.

15.

16.

14 48 50

10 26 28

9 12 15

10 12 16

=h2 3

3=b 3 2

=c 3.5=d 2.8

tanA =7

24

sinA =7

25

cosA =2425

tan12° = 0.21

cos77° = 0.22

sin49° = 0.75

=x 4.8

=A 44° © 2019 M cGr aw - H i l l Educat i on. Al l R i ghts R eser ved.C hapter 8 Pr ac t i ce T es t #1 Page 20 / 23

16.

17.

18. square units

19.

20.

21. Part 1: In , and are

- complementary.

Part 2: Use , , and to fill in the blanks.Make sure to use the appropriate upper-case or lower-case letters.

Part 3: Select all of the true statements.

None of the above is true.

Part 4: Fill in the blank.

22.

=A 44°=b 29.0=c 40.3

=x 52.1°

131.1

=y −14

=v 14

XYZ ∠X ∠Y

x y z

=sinXx

z=sinY

y

z

=cosXy

z=cosY

x

z

=sinX sinY

=sinX cosY

=cosX cosY

=cosX sinY

=cos 57° sin 33°

102.6ft © 2019 M cGr aw - H i l l Educat i on. Al l R i ghts R eser ved.C hapter 8 Pr ac t i ce T es t #1 Page 21 / 23

22.

23.

24. meters

25. , ,

26. feet

27. , , ,

or , ,

28.

29. , ,

30. feet

102.6ft

=x 49.6°

281.4

=C 41° =a 4.3 =b 2.4

69.6

=B 41.4° =A 108.6° =a 64.5=B 138.6° =A 11.4° =a 13.5

b

B

a

A c

C

h

Part 1: Use trigonometry to fill in the blanks.

=sin B =h

cand sin C

h

b

Part 2: Rewrite the equations from Part 1.

=h =⋅c sin B and h ⋅b sin C

Part 3: Use the equations from Part 2 to write an equation

relating and .sinB sinC

=⋅c sin B ⋅b sin C

Part 4: Rewrite the equation from Part 3.

=sin B

b

sin C

c

=A 90.1° =B 72.1° =C 17.8°

57.2 © 2019 M cGr aw - H i l l Educat i on. Al l R i ghts R eser ved.C hapter 8 Pr ac t i ce T es t #1 Page 22 / 23

30. feet

31.

Part 1: Use the Pythagorean Theorem to find .

Part 2: Use the Pythagorean Theorem to find .

Part 3: Use the answer from Part 2 to fill in the blanks.

Part 4: Use the answers from Parts 1 and 3 to fill in the blanks.

Part 5: Use trigonometry to fill in the blank.

Part 6: Use the answers from Parts 4 and 5 to fill in the blanks.

57.2

a2

=a2 +−b x 2 h2 =a2 +−c x 2 h2

=a2 +b2 h2 =a2 +x2 h2

c2

=c2 +−b x 2 h2 =c2 ++x b 2 h2

=c2 +x2 h2 =c2 +−a x 2 h2

=c2 +x2 +h2 −b2 ·2 ·b x

=c2 +a2 −b2 ·2 ·b x

=x a cosC

=c2 +a2 −b2 ·2 ·b a cosC

A

a

B

b

C

c

D

h

x

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