assignment sheet - pingry school
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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