1

Assignment Sheet

1. Quadratic Equation Assignment (in packet)

2. Ratios/Similarity of Polygons Page 244: 25, 29, 31 (drawing not required)

Page 251: 26, 27, 31, 33

3. Similar Triangles Page 257-9: 1-9, 11, 13, 15, 18, 25

4. Similar Triangle Proofs Page 259: 27, 28

Page 266: 1-8

5. Proportional Lengths Page 272-3: 1-3, 5, 7, 10, 11, 15, 21, 23

6. Review Sheet: 1-10, 18 (skip 1f)

7. Test

2

Quadratic Equations

Factor Completely

(1) 224x 17x 20− − (5)

224x 55x + 25−

(2) 224x 2x 15− − (6)

220x 9x 20− −

(3) 224x + 26x 15− (7)

224x + 22x 30−

(4) 224x 11x 13− − (8)

3 224x 26x 44x− −

Solve for x.

(1) 224x 2x 7 = 0− − (5)

3x 1 x + 1 =

x + 3 x

−

(2) 212x + 8x 15 = 0− (6)

13x 1 x + 7 =

2x + 1 2x

−

(3) 212x 19x 18 = 0− − (7)

2

2

7x + 3 3x 4 2x 4 + =

x + 1 x 1 x 1

− −− −

(4) 4x 6 4x + 9

= x 1 2x 7

− −− −

(8) 3 218x + 27x 8x 12 = 0− −

3

Answers

Factor Completely

(1) ( )( )8x + 5 3x 4− (5) ( )( )8x 5 3x 5− −

(2) ( )( )4x + 3 6x 5− (6) ( )( )5x + 4 4x 5−

(3) ( )( )2x + 3 12x 5− (7) ( )( )2 3x + 5 4x 3−

(4) ( )( )24x + 13 x 1− (8) ( )( )2x 12x + 11 x 2−

Solve for x.

(1) 712 12 , − (5) 1

2 , 3−

(2) 5 36 2 , − (6) 7

24 , 1−

(3) 9 24 3 , − (7) 3

8−

(4) 1712 , 3− (8) 32

3 2 , ± −

4

Assignment #1

(1) 0 1 x64 2 =− (8) 0 30x 17x x16 23 =−−

(2) 0 40 3x x2 =−− (9) 0 24 55x x24 2 =−+

(3) 0 20 17x x24 2 =−− (10) ( ) 16 5 x2 2 =−

(4) 0 15 16x x48 2 =−− (11) 2 x

3 x

1 x

1 3x

−+

=++

(5) 0 18 59x x48 2 =+−

(6) ( )( ) ( ) 1 2x 2 x 3x 1 x1 x +=−−−+

(7) ( ) ( ) ( )2 2 23x 2 2x 3 4 x 5x 18+ − − = + +

5

Answers

(1) 8

1± (8) 0 ,

16

15− , 2

(2) −5 , 8 (9) 3

8− ,

8

3

(3) 8

5− ,

3

4 (10)

2

1 ,

2

9

(4) 12

5− ,

4

3 (11) 5 ,

2

1−

(5) 16

9 ,

3

2

(6) 1

(7) −11 , 7

6

Ratios/Proportions

1. Ratios MUST be reduced!!!

2. Ratios can be written as 1) words ____________

2) colon form ____________

3) fraction form ____________

3. Ratios must be in the same unit. (i.e. hours to hours, minutes to minutes etc)

Compare 7 inches to 2 feet

4. Order matters on ratios!

A. Sally has attempted 48 shots and made 36. What is the ratio of shots made to shots attempted?

B. A telephone pole 7 meters tall snaps into two parts. The ratio of the two parts is 3 to 2. Find the length of each part.

ABCD is a parallelogram. Find each ratio.

C) AB: BC __________

D) BC: AD __________

E) m∠A: m∠C __________

F) AB: perimeter of ABCD __________

G. The measures of the angles of a triangle are in the ratio of 3:4:5. Find the measures of each angle.

A B

D

6

10 C

7

A proportion is a set of 2 equal ratios, such as 1

3 and

12

36

a c =

b d

Solve for x:

1. 15 5

x 7= 2.

x x 4

x 5 x

−=

+ 3.

x 1 10

x 2 3x 2

−=

+ −

4. If x

7 =

3

2 then

x + 7

7 = ________

Similar Polygons

Two polygons are similar if their vertices can be paired so that:

Corresponding angles are congruent AND corresponding sides are in proportion.

Corresponding vertices must be listed in order:

Given polygon ABCDE ~ polygon VWXYZ

1. Congruent angles.

2. Proportions of sides.

Scale Factor:

8

1. Given quad ABCD ~ quad A’B’C’D’

Find x, y, and z

What is the scale factor? __________

THE RATIO OF THE PERIMETERS OF SIMILAR FIGURES = _________________________

Draw two polygons with congruent angles that are not similar

Can you draw two triangles with congruent angles that are not similar?

A’ B’

C’ D’

30

50

22

y

A

C D 12

30

x z

B

9 AA Postulate:

Tell whether or not the following triangles are similar.

1. 2. 3. 4.

5. Find the value of x 6. Find x and y

7. Given: AC || BD

Prove: CO OA

DO OB=

3 5

1 x 3

4 3

x

y

8

A

O

C

D

B

10

8. Given: KH is the altitude to hypotenuse GJ of △GHJ

Prove: KH HJ KJ

HG GJ HJ= =

Then: HJ x HJ = GJ x KJ

9. Given: ABCD is a parallelogram

Prove: BF FE

CF FD=

Then: BF FD = CF FE

1

3

2

4 G

K

J

H

B A E

D C

5

4 3

2

1 F

11

10. Given: <MET ≅ <RST ST MR⊥

Prove: ME ST = RS ET

11. Given: △MTR is isosceles with legs MT and RT

,VO MR SP MR⊥ ⊥

Prove: MO RS = RP MV

M

T

R

E

S

1 2

3

4

O P

4 3 R

S

T

V

M 2 1

12

p. 259 # 34

SQUARE ABCD FIND: HX, HY, HW, BF, FC, CG, DE, EA, EH, HF

AB = 16

DG = 12

AH = 10

HG = 10

A

C

B

D

E

F

G

H Y Z

X

W

16

10

12

13

O, F, H, K are midpoints . 3

4

OF

OK= MG + EJ = 42

Find OK

O

E F G

H

J K M

14 Theorems for Similar Triangles.

1. The measures of the sides of ∆ABC are 4, 5, and 7. The measures of the sides of ∆XYZ are 16, 20 and 28. Are the triangle similar? If so, what is the scale factor?

2. In ∆ABC AB=2, BC=5 and AC=6. In ∆XYZ XY=2.5, YZ=2, and XZ=3. Is ∆ABC ~ ∆XYZ?

3. Name the similar triangles and give the postulate or theorem that justifies your answer.

a) b) c)

4. Given ∠B ≅ ∠ DEC Prove: ∆ABC ~ ∆DEC

A

C B

D E

A

C B

D E

10

5

6

3

A

B

D

C E

5

10

6

8

12

8

15

PROVE: AB x ED = CD x BE

5)

1. Trapezoid ABCD 1. Given

bases AB, DC

6)

Prove: ||HM JK

1. GJ GK

HK GM= 1 G≅∢ ∢ 1. Given

A B

C D

E

1 3

4 2

J

H M

G

K 1

2 3

4

16

Proportional Lengths

A line that intersects two sides of a triangle and is parallel to the third side intersects the sides

Lets do some sample problems

1. Given the picture to the right:

a) CD

DA=

b) If CD = 3, DA = 6 and DE = 3.5, then AB = ______.

c) If CB = 12, EB = 8 and CD = 6 then DA = ______.

d) If CD = 1

2, DA =

1

3 and EC =

3

4 then BC = ________

A B

C

E

d

c

a

b

D

17 2. Given the drawing,

a) Write an acceptable proportion.

b) If a = 2, b = 3 and c = 5 then d = ______.

c) If a = 4, b = 8 and c = 5 then c + d = ______.

Triangle Angle-Bisector Theorem: If a ray bisects an angle of a triangle, then it

______________ the _________________________ into ___________________________

to the ____________________________,

Again, lets draw a picture to show what this means:�

3. Find the value of x: 4. Find the value of x:

a

b d

c

5

8

x

2

10 12

24

x

18

Review Sheet Ch 7 Similarity

1. Refer to the figure, given DE BC

a) AD = 7, BD = 3, DE = 6 Find BC _________

b) AD = 3, BD = 5, AE = 4, Find CE _________

c) AD = 4, AB = 10, BC = 25 Find DE _________

d) AD = CE, BD = 4, AE = 9 Find AB _________

e) AD = x-1, BD = 5, AE = 1, CE = x+3, DE = 2x+1

Find BC _________

** EXTRA HARD** f) AD = 2x, BD = x+3, AE = 4x-1, CE = 5x, BC = 6x+2

Find DE _______

2. Refer to the figure, given ∠1≅∠2

a) AC = 6, BC = 8, BD = 5 Find AD __________

b) AB = 10, AC = 4, BC = 8 Find AD __________

c) AC = 3, AD = x-4, BC = x, BD = 4 Find BC ________

3. 4.

x = _________ ABCD is a parallelogram, sides as marked

BE _______ CE ________ CF _________

5. Given: AB || CD

Prove: AE CE = DE EB

A

B C

D E

B

C D F

A

E

12

4

10 8 7 x+2

3 x

A B

C D

E

1 2

3 4

A B

C

D

2 1

19

6. 7.

Given AB || CE || FG Given AB || CE, ∠1≅∠2

Find AD ______ BE ______ FG ______ Find AB ______ AD _______ AE _______

DE ______

8. Given: ∠1≅∠2, sides as marked 9. Given BD || AE, DF || AC, sides as marked Find: AC _______ BD _______ Find: AC ______ BD _______ CD ________

10. Given BE and CD are altitudes

Prove: AE AC = AD AB

C

A B

E D

F G

6

8

5

7 9

C B

A

2

6

1

E

D 10 8

A

B

C D

12 10

14

1 2

A

B

F

E D C

5

8

6

4

1

E D

C B

A

2

3

4

20

11)

x = ___________

y = ___________

z = ___________

** FIGURE NOT DRAWN TO SCALE **

12) a = ___________

b = ___________

z = ___________

4 a

6 7 b

5

x

y

z

3

6

16

5

z

7

21 13)

Prove: BF x ED = CE x AE

1. || , ||FC AD CE AB 1. Given

14) Prove: ( )2ST TW x RT=

1. ,RS ST SW RT⊥ ⊥ 1. Given

C B

A

2 1

E

D

3 4

5

6 7 F

R

W

T S 2

1

3 4

22

15) Find the ratio of x to y 8 7

2 3 6 4x y x y=

− −

ABC DEF△ ∼△ <A = 50

<D = 2x+5y

16) <E = 5x + y

<B = 94 - x

Find <F = __________

17) Find the perimeter of ABC△ ___________

18) Find AD, AB, AC

A B

C

D E

F

A

B C

x-2

9

4

5

y

x+3

A B

C

D

4

16

23

SIMILARITY PROBLEMS

1) Given a square in a right triangle. Find the length of its side given the information in the picture.

2)

x

12 5

Given: AF FC, FC BE AND DC⊥ ⊥

Find : DC

A

B

C

D

E F

15

9

3

24 3) Given the diagram, find BD, CD, AC

4)

Given: the figure as marked. Show <BAC = <CAD

15 16

A

B

C

D

12 18

8

27

12

A

B C

D

25 Extra Practice Problems

(1) Given the figure to the right DE BC� .

(a) AD = 6 , BD = 4 , DE = 5 , CE = 6

Find: BC________ , AE________

(b) AD = CE = x + 1 , BD = x – 1 , AE = 2x – 1 , BC = 2x

Find: x________ , DE________

(2) (3)

Given: ∠1 ≅ ∠2, sides as marked Given: DE AC , EF AB� � ,

sides as marked

Find: BD________ , AD_______, Find: BD____, CF____, DE____, EF____

(4) (5)

Given: ∠1 ≅ ∠2 ≅ ∠3, sides as marked Given: AB AC , DE BC⊥ ⊥

Find: AC_____ , CD______, DE______ Prove: BC DE = CE ABi i

A

B C

D E

A

B C

D

E

F 15

18

4 5

A

B C

D

E

1 2

3

6 9

10

A

B C D

E

1

2 3

4

5 6

A

B

C

D

1

2

21

9

16

26

Answers

(1) (a) 253

BC = , AE = 9

(b) x = 5 , DE = 6

(2) 48 277 7

BD = , AD =

(3) 20 253 3

BD = , CF = 10 , DE = 8 , EF =

(4) AC = 15 , CD = 9 = DE = 6