To take the exam, you can either print it out and do your work on the printed paper, or you can do the work

in your own notebook. If you cannot print the exam, you have to create your own triangle 4ABC for Problem

8. It doesn’t have to be exactly the same triangle as on the exam, but it has to be acute and scalene.

final_exam__1_.pdf

Unformatted Attachment Preview

FINAL EXAM

MAT 040

May 6, 2020

DUE: FRIDAY, MAY 8, 2020, 11:59 PM

Dr. Anke Walz

In order to receive full credit, you must show all of your work and make your methods clear. Please

write neatly and legibly. Constructions are to be done with compass, straightedge, and pencil, and should be

very neat and precise!

Submit your work on D2L in the appropriate Assignment folder. The deadline is Friday, May 8, 2020, at

11:59 pm. You will be able to submit your work after the deadline, but you will not get full credit for it.

To take the exam, you can either print it out and do your work on the printed paper, or you can do the work

in your own notebook. If you cannot print the exam, you have to create your own triangle 4ABC for Problem

8. It doesn’t have to be exactly the same triangle as on the exam, but it has to be acute and scalene.

You can use your notes and the materials I posted on D2L, but you cannot talk to each other or use other

resources. You can talk to me if you have questions. Don’t cheat!

To submit your finished exam on D2L:

1. Either scan your exam or take VERY CLEAR pictures (in the correct orientation, not sideways!) Check

the quality of the pictures before you upload them and make sure that your work is legible and recognizable,

and that the problems are in the correct order and clearly labeled.

2. If you submit your work in a single file, name it “Final Exam”.

3. If you submit your work in several files, name them “Problem 1”, “Problem 2”, etc., or “Page 1”, “Page

2”, etc.

4. Do NOT submit any files with an .heic extension, I can’t open them. Convert your file to a .pdf or a .docx

file.

5. Do NOT submit your work as a .zip file.

6. Do absolutely NOT email your exam to me. You have to submit it on D2L. The submission folder will

remain open after the deadline, but you will lose points for submitting your work late.

If you choose not to follow these instructions, you will lose points on your exam.

Problem 1 (40 points) For each of the following statements decide whether it is true or false. You do not

have to justify your answers.

(1) The orthocenter of a triangle 4ABC has the same distance from

all the vertices of the triangle.

Answer:

(2) Every triangle has at most one right interior angle.

Answer:

(4) Every triangle has at least two obtuse exterior angles.

Answer:

(5) It is impossible to construct a regular nonagon with compass

and straightedge.

Answer:

(6) Points that lie on a single line are called commingled.

Answer:

(7) If a, b, c satisfy the inequality a + b > c, then a, b, c are the sides

of a triangle.

Answer:

(8) The two acute interior angles of a right triangle are complementary.

Answer:

(9) The interior angles of a triangle are supplementary.

Answer:

(10) If two angles are complementary, then both angles must be acute.

Answer:

(11) Two intersecting lines form two linear pairs of angles.

Answer:

(12) Corresponding angles are supplementary.

Answer:

(13) If two angles form a linear pair, then they are supplementary.

Answer:

(14) If ∠A and ∠B are vertical angles, and ∠B and ∠C are alternating

exterior angles, then ∠A ∼

= ∠C.

Answer:

(15) Let P and Q be regular polygons such that P has half as many sides as Q.

Then the interior angles of P are half as large as the interior angles of Q.

Answer:

(16) The sum of the interior angles of a hexagon is 750◦ .

Answer:

(17) If the sum of the measures of the interior angles of a polygon P is 2700◦ ,

then P has 17 sides.

Answer:

(18) If each interior angle of a polygon P measures 135◦ , then P is a regular octagon.

Answer:

(19) All equiangular polygons are equilateral.

Answer:

(20) Without undefined terms Geometry would be pointless.

Answer:

Problem 2 (30 points) Consider the information given in the following figure (which is not drawn to scale):

∠BAG ∼

Gr

= ∠ABG

A

@

AG ⊥ DH

3 A@

b

m(∠a) = m(∠ABG) = 70◦

H r 4 A @

A @

HH

HH5A

m(∠b) = m(∠EGF ) = 25◦

@

HAH

r6 @

F A HH @

7

H

@ E

A

H

r

H

@

A

8 HH

A

@ H

@ HH

A

HH

@

A

HH

@

A

H

@

A

HH

@

A

H

HH

@

A

a

1

9

2 HHr

10

@r

Ar

r

A

B

C

D

Determine the measures of the angles labeled 1, 2, 3, . . . , 10. You absolutely have to justify your answers,

using correct terminology! A correct answer without sufficient justification will only receive partial credit.

Problem 3 (20 points) Give a complete classification for each triangle. If the triangle does not exist, explain

why. You have to justify your answers!

(a) a = 2, b = 3, c = 5.

(b) a = 5, b = 11, c = 10.

(c) a = b = 5, γ = 120◦ .

(d) a = 3, b = 4, c = 5, α̂ = 90◦ .

Problem 4 (10 points) Refer to the illustration given below to answer all the multiple-choice questions.

Mark your answer clearly! You do not have to justify your answers!

sC

A

A

A

Y s

A

@

A

@

A

@

Gs

s R A

@

HH

A

@

@

H

A

@ HHs

@

A

@

H

@

E @s T HH

A

@

H

HH @ A

H @A

HHA

H

s

s

@As

O

M

B

s

A

(1) M is called

(A) the footpoint of the altitude CM

(B) the midpoint of the side AB

(C) the median of the side AB

(2) BY is called

(A) the median of AC

(B) the perpendicular bisector of AC

(C) the altitude of AC

(3) T is the

(A) center of the circumcircle of 4ABC

(B) orthocenter of 4ABC

(C) centroid of 4ABC

(4) The orthocenter of 4ABC is

(A) C

(B) E

(C) R

(5) The following points lie on the circumcircle of 4AM C:

(A) A, B, C

(B) A, M, Y

(C) A, C, M

AO ∼

= OB, AG ∼

= GC

AC ⊥ GE, AC ⊥ Y B

AB ⊥ OE, AB ⊥ M C

←→

Problem 5 (20 points) Given is a line AB. Use compass and straightedge to construct a point X such that

m(∠BAX) = 52.5◦

Clearly label the point X, and outline the angle ∠BAX!

r

A

r

B

–

Problem 6 (20 points) Use compass and straightedge to construct a regular octagon.

Clearly outline the octagon!

Problem 7 (20 points) Given is a segment AB.

Use compass and straightedge to construct a Golden Triangle with base AB.

Clearly labels the third vertex of the triangle, and clearly outline the triangle!

r

A

r

B

Problem 8 (20 points) Given is an acute, scalene triangle 4ABC.

Use compass and straightedge to construct its orthocenter H and its centroid G.

Draw the Euler line l.

Clearly label the orthocenter H, the centroid G, and the Euler line l!

C

r

C

C

C

C

C

C

C

C

C

C

C

C

C

r

A

C

C

Cr

B

Problem 9 (20 points) Given is a segment AB.

Use compass, straightedge, and pencil, to construct a point C such that 4ABC is an equilateral triangle.

Also construct the circumcenter O and the circumcircle CO of 4ABC.

Clearly label the vertex C, the circumcenter O, and the circumcircle CO !

r

r

A

B

…

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