level 1 geometry final exam june 22, 2006 · level 1 geometry final exam june 22, 2006 name teacher...

Level 1 Geometry

Final Exam

June 22, 2006

Name

Teacher

Block

The last page is a formula sheet which can be torn off.

PART I: PLEASE SHOW ALL OF YOUR WORK! If no work is shown, then credit maynot be given. If you cannot fit your work in the space given, please use extra paper andremember to put your name on it. Neatness is expected. If I cannot read an answer, Icannot correct it and it will be marked as incorrect. Answers will be accepted in anycorrect form (either in simplified radical form or decimal.) Don’t forget to include unitsin your answers. If you do not understand the phrasing of a question please ask. Goodluck!

1. (3 points) Six times an angle equals twice its complement. Find the angle.

2. (3 points) A cylinder has a height equal to the diameter of the base. The volumeof the cylinder is 500π cubic centimeters. What is the height of the cylinder?

3. (3 points)A regular square pyramid fits exactlyin a cube with sides of length 10.

That is, the pyramid and cube sharethe same base EFGH, and vertex Plies in the plane ABCD.

What is the volume of the portion ofthe cube which is not occupied by thepyramid?

P

H G

FE

D C

BA

_____________________

4. (5 points) If k is parallel to m, find the measures of angles a, b, c, d, and e.

m

k

58°

91°

e

d

cb

a

a =

b =

c =

d =

e =

5. (2 points) Determine the coordinates of point M such that AM is the median to BC.

C (13, -6)

B(3, 4)

A (9, 10)

6. (3 points) Find the measure of ∠ACB.

4x - 20

x + 4

2x

B

A DC

7. (4 points) Find the length of base AB of trapezoid ABCD.

60 °45 °

10 cm

8 cm

A B

CD

8. (3 points) If ABCDE is a regular polygon, find the measure of ∠F.

E

D

C

FBA

9. (3 points) Two ladders are leaned against a wall such that they make the same anglewith the ground. The 10’ ladder reaches 8’ up the wall. How much further up the wall(x) does the 18’ ladder reach?

10'

18'w a l l

x

10. (3 points) Determine the area of the regular hexagon below given that the length ofOP is 12 cm.

P

C

D

F

E

A

O

B

A

BC

DE

FG

11. (3 points) Determine the length of the lateral edge ( AC ) of the regular squarepyramid given the length of a side of the base is 12 cm. and the length of the altitude( AF ) = 8 cm.

12. (4 points) A, B, C and D are points on a circle. AB = CD. m∠ABC = 22º, and m∠BCD =67º. Find the size in degrees of minor arc AB.

67º

22º

C

A

D

B

_____________________

13. (3 points) PT is tangent to circle O at T. The diameter of the circleis 16. PT = 15. Find PX.

15

XT

P

O

14. (3 points)Two models of the same sailboat are made. Each one is similar to the original, so theyare similar to each other. The larger model is 120 cm long and the smaller is 30 cmlong.

a. How does the height of the larger boat compare to the height of the smaller?____________________________________________________________

b. How does the area of the sails of the larger boat compare to the area of the sailsof the smaller?____________________________________________________________

c. How does the volume of the hull (the body) of the larger boat compare to thevolume of the hull of the smaller?______________________________________________________________

30 CM 120 CM

15. (4 points) A person looks out from an upper floor of a building across a level 75foot wide street to another building. By looking up at an angle of elevation of 81º shesees the top of the building and by looking down at an angle of depression of 35º shesees the base of the building. How tall is the building she is looking at?

35 °81 °

16. (3 points) Quadrilateral GEOM has vertices G(-3, -6), E(2, -4), O(1, 3), and M(-4, 1).Show that GEOM is a parallelogram.

17. (2 points) One or more transformations moves triangle ABC to triangle A'B'C' asshown. Which of the following describes the transformations?

I. A rotation.II. Two reflections.III. A reflection followed by a translation.

Circle the correct answer.a. I only. b. II only. c. III only. d. I and II only. e. I and III only.

6

4

2

-2

-4

-6

-5 5

C'B'

A'

C B

A

18. (3 points) Assume that the following are true statements without exception:

All Americans know who Abe Lincoln was.Anyone who does not know who Abe Lincoln was is not a history buff.

Circle which of the following is definitely true or not necessarily true.

a. All Americans are history buffs. Definitely Not necessarilytrue true.

b. If Jose is not an American, he does not Definitely Not necessarily know who Abe Lincoln was. true. true

c. All people who are history buffs Definitely Not necessarily know who Abe Lincoln was. true true.

PART II: Do 2 of the following 4 proofs. Indicate clearly which twoare to be graded.

19. (5 points) Given: ADAB ≅ BCAB ≅ ABCBAD ∠≅∠

Prove:

€

ΔXCD is isosceles.

X

D C

BA

20. (5 points)

Given: ABCD is a parallelogram. DQBP ≅

Prove: X is the midpoint of

€

AC . Q

P

X

D

C

B

A

21. (5 points)

Given: ABDF is a parallelogram

Prove: FEBD

DFCB

=

F

E

D

C

B

A

22. (5 points) A coordinate proof.

Given any triangle PQR positioned in acoordinate plane as shown, so that P = (0, 0), Q = (a, 0) and R = (b, c).

Prove that the segment

€

MNconnecting themidpoints of

€

RPand

€

RQ is parallel to

€

PQand half the length of

€

PQ .

NM

R (b, c)

Q (a, 0)P (0, 0)

y

x

Formula sheet for Level 1 Geometry classes, June 2006 final exam.

Pythagorean triples. 3-4-5 5-12-13 7-24-25 8-15-17 9-40-41

Trigonometry.

Adjacent

HypotenuseOpposite

x

€

s =oh

c =ah

t =oa

sin(x) =oppositehypotenuse

cos(x) =adjacenthypotenuse

tan(x) =oppositeadjacent

Special right triangles.

x 2

x

x

x 3

2xx

45º

45º

60º

30º

Coordinate geometry. Given points

€

A(x1,y1) and

€

B(x2,y2):

€

AB = (x2 − x1)2 + (y2 − y1)

2

Midpoint of

€

AB____

=x1 + x22

, y1 + y22

Slope of

€

AB =riserun

=y2 − y1x2 − x1

Circumference / Area / Volume.Circle:

€

C = 2πrA = πr2

Triangle:

€

A = 12 bh

Quadrilaterals:Parallelogram:

€

A = bhSquare:

€

A = s2Trapezoid:

€

A = 12 h(b1 + b2)

Rectangle:

€

A = lwQuadrilateral with

€

⊥ diagonals:

€

A = 12 d1d2

Cylinder:

€

V = πr2hCone:

€

V = 13 πr

2hPyramid:

€

V = 13 base area ⋅ h

Sphere:

€

V = 43 πr

3

level 1 geometry final exam june 22, 2006 · level 1 geometry final exam june 22, 2006 name teacher...

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