West Springfield High School

Geometry

2008-9

1. My dad was going to cut down a dead tree in our yard one day, but he was afraid it might hit some nearby power lines. He knew that if the tree were over 45 feet tall, the tree would hit the power lines. He stood 30 feet from the base of the tree and held a ruler 6 inches in front of his eye. He lined the bottom of the ruler up with the base of the tree, and saw that the top of the tree lined up with a point 8 inches high on the ruler. He then knew he could safely cut the tree down. How did he know? 2. In the figure, MN ! OP , OP=12, MO=10, and LM=5. Find MN. P W

L

5 P X 3

12

N

5 M 10

O

Y

10

12

3. The lengths in the diagram are as marked, and WX ! YZ . PY and WX.

A

4. Find BC and DC given AD=3, BD=4 and AB=5.

A

Y

3D

5

Find

Z

E

D

4

X

B

C

B

C

5. Given that DE ! BC and AY ! XC , prove (EY/EX)=(AD/DB). 6. In the figure we have (AB/AC)=(AD/AE)=4/5, and clearly ∠BAD=∠CAE. We wish to prove that ∆ABD≅∆ACE. (Note that we cannot assume that BD ! CE . We have to prove it!) a) Suppose we draw a line through B parallel to CE that hits AE at X as in the diagram. What do we know about ∆ABX and ∆ACE? b) Given that AE=15 in both diagrams, what is AX? c) What can we conclude about D and X? d) What can we conclude about ∆ABD and ∆ACE? e) What similarity rule can we create from this investigation?

A 8 B 2 C

A

B

8

12 D

B 3 2 E C

6

D

5

E

A

4

C

7. Given AC=4, CD=5, AB=6 as shown, find BC if the perimeter of ∆BDC is 20.

53

D

West Springfield High School

Geometry

2008-9

1. Given the side lengths shown in the diagram, prove that AE ! BC and AB ! DE .

A

4 12

A

5

B 6

z D

C

36 27 F y 64 x

D 10

E

E

45

B

C

2. In the diagram, DE ! BC , and the segments have the lengths shown in the diagram. Find x, y, and z. 3. As shown, ∠A=90°, and ADEF is a square. Given that AB=6 and AC=10, find AD.

B D

P

E Q

A

F

X

R

C

4. In the diagram, PX is the altitude from right angle ∠QPR of right triangle ∆PQR as shown. Shown that PX2=(QX)(RX), PR2=(RX)(RQ) and PQ2=(QX)(QR) 5. ∆ABC ≅∆XYZ, AB/XY=4, and [ABC]=64. In this problem we will find [XYZ]. a) Let hc be the altitude of ∆ABC to AB and let hZbet the altitude of ∆XYZ to XY. What is hc/hz? b) Find [XYZ]. c) What general statement about the areas of similar triangles can you make? 6. In the diagram, ∠ACQ=∠QCB, AQ ^ CQ , and P is the midpoint of AB . Prove PQ ! BC .

A P

7. Flagpole CD is 12 feet tall. Flagpole AB is 9 feet tall. B Both flagpoles are perpendicular to the ground. A straight wire is attached from B to D and another from A to C. The flagpoles are 40 feet apart, and the wires cross at E, which is directly above point F on the ground. We wish to find EF. a) Use similar triangles to find ratios of segments that equal EF/AB. b) Use similar triangles to find ratios of segments that equal EF/DC. c) Cleverly choose one ratio from each of the first two parts and add them to get an equation you can solve for EF.

54

Q C

D A 9 B

E F 40

12 C

West Springfield High School

Geometry

2008-9

1. In this problem we will explore why AA Similarity works. Do not use AA similarity to solve the problem! In the diagram, we have two triangles (∆ABE and ∆ACD) with equal angles, and sides with lengths as marked. Our goal in this problem is to find BE and DE, and discover a process to prove that if the angles of one triangle equal those of another, then the corresponding sides of the two triangles are in constant proportion. We will make heavy use of the Same Base / Same Altitude principle. A

a) What are [ABE]/[ACE] and [BEC]/[BED]? b) Use the previous part to show that [ACE]=[ABD]. c) What is [ABE]/[ABD]? d) Use the previous part to find AD. e) What is BE? f) Can we use our work in this problem to porove that if two angles of one triangle equal those of another triangle, then the triangles are similar? 2. a) Find AC and BC

B

C

8 E

B 3 14

C

D

F

Find HJ.

4

E

A 7

b)

6

21

7

G

16

I

9

D H

c) Find ON and MN.

d)

J

Find RS.

O

2 L

1.2 1.6

M

N

3. If two isosceles triangle have vertex angles that have the same measure, are the two triangles similar? Why or why not?

W

X

4. In the diagram, WXYZ is a square. M is the midpoint of YZ , and AB ^ MX . a) b) c) d)

A

Show that WX ! XY . Prove that AZ=YB. Prove that XB=XA. Prove that ∆AZM≅∆MYX, and use this fact to prove AZ=XY/4.

Z

M

Y B

55

West Springfield High School

Geometry

2008-9

5. In triangle ∆ABC, AB=AC, BC=1, and ∠BAC=36°. Let D be the point on side AC such that ∠ABC=∠CBD. B a) Prove that ∆ABC≅∆BCD. b) Find AB.

x

6. Find x in terms of y given the diagram shown.

G

D

A

C

5

5

E

y

F

H

7

7. Find DE in the figure below left. I

A

3

4.5

B

C

D

F

E

6

2

J

M

3 E H

G

8. In the figure above right, M is the midpoint of EH and of FG . E and F are midpoints of IM and MJ respectively. Prove that IJ ! GH . 9. Show that if WX2=(WX)(WY) in the diagram below left, then ∠WZX=∠WYZ.

P

W X

Z

C

Q

R

B

Y

A

10. In the diagram above right, ∠PRQ=∠PQA=90°, QR=QA and ∠QPC=∠RPC. a) Prove ∠QCB=∠QBC. b) Prove RA ! PB . 11. Two isosceles triangles have the same ratio of leg length to base length. Prove that the vertex angles of the two triangles are equal. 12. X and Y are on sides PQ and PR , respectively of ∆PQR such that XY ! QR . Given XY=5, QR=15, and YR=8, find PY.

56

West Springfield High School

Geometry

2008-9

13. In the figure below left, the area of ∆EDC is 25 times the area of ∆BFD. a) Find CD/DB. b) Find [EDC]/[ABC]. c) Find {AFE}/[ABC]. A

W

X

E

D B C

F

Z

B

D

Y

A

C

14. In the diagram above right, WZ ! XY and WX ! YZ . WA and WB hit XZ at C and D, respectively, such that ZC=XD. a) Prove that ZC/XC=AC/WC. b) Prove that XD/ZD=DB/WD. c) Prove CD ! AB . 15. In the diagram below right, PQ=PR, ZX ! QY , QY ^ PR , and PQ is extended to W such that WZ ^ PW . a) Show that ∆QWZ≅∆RXZ. b) Show that YQ=ZX=ZW. R

Y

R

P

A

B

P

X

Y

Q

X

W Q

Z

16. In the diagram above left, PA and BQ bisect angles ∠RPQ and ∠RQP, respectively. Given that RX ^ PA and RY ^ BQ , prove that XY ! PQ .

57

West Springfield High School

Geometry

2008-9

REVIEW PROBLEMS 1. In each of the parts below, either identify all pairs of similar triangles or state that there are not any pairs of triangles that are necessarily similar. For each pair of similar triangles you find, state why the triangles are similar. a)

b)

A

I

E F

D

C

B

H J

G

O

c)

8

d)

K

P

12

M

16

Q

4

N

L

S R

e)

f) Y

W

T

18

9

10

4

2

12

18

V X

U

3. Find x and y in the diagram, given the angle equalities and side lengths shown in ∆PQR and ∆ABC. Q

C

8

6

B

x y

3 A

P

12

R

58

West Springfield High School

Geometry

2008-9

4. Points P and Q are on AB and AC , respectively, such that BC ! PQ . Given AB=12, PB=9, and AC=18, find QA. 5. The side lengths of a triangle are 4 centimeters, 6 centimeters, and 9 centimeters. One of the side lenghs of a similar triangle is 36 centimeters. What s the maximum number of centimeters possible in the perimeter of the second triangle? 6. What’s wrong with the diagram shown below left?

2 A

3.5

D

4.5

D

E

24 E

B

11

13 B

C

6

9

6

A

C

7. Find DE in the figure above right.

W

8. Why is the diagram shown below left impossible? P

X

8 Q

R

12

Y

4

T

6

Z

6

10 S

V

9. Find WY and YV in the figure above right.

10. ∆ABC is equilateral. M is on AB and N is on AC such that BM = CN. a) Prove that AM=AN. A b) Prove that ∆AMN is equilateral. N

M

C

B

11. Given ∆ABC≅∆YZX, [ABC]=40, [YZX]=3360, AB=9, and BC=12, find the following: a) YZ. b) The length of the altitude to side XZ of ∆YZX.

A

E

B

12. Let ABCD be a rectangle as shown below left, with AB=25 and BC=12. Let E be a point on AB such that AE