Download Chap 7 Notes

-1-
Name:_____________________________
Geometry Rules!
Period:_______
Chapter 7 Notes
Notes #36: Solving Ratios and Proportions and Similar Triangles (Sections 7.1 and
7.2)
3
, 3 to 4, 3:4
4
3 6

Proportion: two ratios that are equal to each other.
4 8
Ratio: a comparison of two quantities.
A. Solving Proportions
 Cross-multiply and set the products equal to each other
 Be sure to use FOIL when you are multiplying binomials together
 Solve for the variable
Solve for x:
1.)
2 3

x 12
2.)
3.)
x3 4

2
3
4.)
x 7

3 5
x 1 x  5

2x  3
2x
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B. Properties of Proportions
Complete:
5.) If
x 3
 , then 7 x  ____
2 7
7.) If a : 2  5 : 3, then 3a  ____
9.) If
x y
x
 , then  ____
5 2
y
11.) For the given figure, it is given that:
KR = 6, KT = 10, KS = 8
6.) If
5 3
 , then 3x  ____
x 2
8.) If
x 2
y
 , then  ____
y 9
x
10.) If
x y
x3
 , then
 ____
3 4
3
KR KS

. Solve for the missing lengths.
RT SU
RT = ____
K
SU = ____
R
S
T
KU = ____
U
C. Similar Polygons
Similar polygons have the same _________ but not necessarily the same ________.
Example of similar triangles:
B
Y
6
A
3
10
X
8
5
4
Z
C
 Their corresponding angles are ___________________
 Their corresponding sides are in a ______________ _______________________
 This ratio is called a ____________ ______________ and in this case is _______
 We show that they are similar with this statement: ___________________
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12.) ABCDE
A' B ' C ' D ' E '
a) scale factor = _______
9
C
x 160
D
b) mA '  _____ , mD  _____
mC '  _____
B
6
100
A
C'
y
30
2
c) x = _____, y = ______, z = ______
E
8k
D'
B'
4
A'
E'
z
The figures are similar. Solve for the variables. (Hint: redraw the diagram as two
figures)
13.)
18
12
10
y
x
z
16
8
14.)
12
16
x
y
9
24
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D. Algebra Practice:
Factor:
15.) 4 p 4 r 3s 2  16 p 2 r 3s 4
16.) 3x 2  15 x  21
17.) 2 x3  8 x 2  8 x
Simplify:
18.) 3 15
19.) 2 8 3 12
20.)  3 5 
22.) 2 x 2  10 x  28
23.) 12 x 2  7 x  10
2
Solve:
21.) 3x 2  5 x  2  0
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Notes #37: Similar Triangles (Sections 7.3 and 7.5)
Similar triangles have:
 ________________ corresponding angles
 sides that are in __________________ _______________________
You can conclude that two triangles are similar if:
_________: two pairs of corresponding angles are congruent
_________: all three pairs of sides are in the same proportion
_________: two pairs of sides are the same proportion and their included angles are
congruent
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Are the triangles similar? If so, state the similarity and the postulate you used.
 Re-draw the triangles in matching positions
 Mark congruent angles
 Test sides for a constant proportion:
small medium l arg e


small medium l arg e
 Look for these patterns: AA~, SSS~, SAS~
1.)
2.)
F
B
35
60
O
D
85
E
N
P
60
A
C
Q
M
3.)
4.)
B
F
E
15
4
4
6
6
8
B
F
D
C
10
D
12
A
C
A
5.)
6.)
R
5
4
8
10
Q
S
6
70
9
X
10
70
Y
7.5
6
Z
15
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State whether the figures are always, sometimes, or never similar:
 do they always, sometimes, or never have the exact same shape?
7.) two squares
8.) two congruent triangles
9.) two rectangles
10.) two rhombuses
11.) two pentagons
12.) two regular octagons
Proportional Lengths (Section 7.5)
A. Triangle Proportionality
 A parallel slice cuts a triangle’s sides proportionally ( Side-Splitter Theorem)
a
j
c
k
,
a

c
a

c
,
a

j
d
b
Example: Solve for x
12
33
a

b
20
x
d

k
,
j

b
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B. Angle Bisector Proportionality
 An angle bisector proportionally divides the opposite side
z
w
y
z

y
w

z
x
x

w
Example: Solve for x:
x
21
8
14
C. Parallel Line Proportionality
 Parallel lines proportionally divide their transversals
a
a

b
b
d
c
d

c
Example: Solve for x:
18
9
24 - x
x
a

c
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Solve for the variables:
1.)
2.)
14
9
6
24
18
4
x
x
60
3.)
4.)
20
x
x
12
18
21
x+ 2
5.)
6.)
3
16
12
12
2x
7.5
4
25
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13.) Write the equation of a line that
contains the point ( -4, 3) and has a slope
of  .
15.) Write the equation of a line in
standard form that is parallel to
3 x  6 y  5 and contains the point ( 1, 4).
16.) Write the equation of a line that
contains ( -2, -3) and ( 4, -9) in standard
form.
18.) Write the equation of a line that is
perpendicular to y  2 x  5 and contains
the point ( -6, 7)
1
2
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Notes #38: Similarity in Right triangles ( 7.4)
Geometric Means and Similar Right Triangles
A. Geometric Mean
asks the question: “what number, squared, equals the product of two given numbers?”
Find the geometric mean of the listed numbers:
 Use the given numbers in this equation: x2 = ab
 Solve for x
1.) 9 and 16
2.) 12 and 3
3.) 5 and 15
B. Similar Right Triangles
 When an altitude of a right triangle is drawn to its hypotenuse, three similar right
triangles are formed:
y
x
a
y  a(a  b)
z  b(a  b)
x  (a)(b)
z
b
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Solve for the variables:
 Re-draw the three triangles and label all sides
 Set up proportions to solve for the variables
 Look for ways to use the Pythagorean theorem
4.)
p
n
m
5
20
m  (5)(25)
p  (20)(25)
n  (5)(20)
5.)
1
4
a
1
9
b
c
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6.)
3
5
y
x
z
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Notes #39—Section 8.1
Pythagorean Theorem
In Words:
Pictures/Symbols:
Example:
Find the missing side of the
triangles below.
1.)
In a _____________
triangle, the sum of the
__________ of the lengths
of the _________ is equal
to the __________ of the
length of the
______________.
x
5
12
2.)
5
x
3
A
Pythagorean Theorem.
is a set of whole numbers a, b, and c, that satisfy the
Examples: Do the lengths of the sides given form a Pythagorean triple?
3.) 8, 15, 17
4.) 7, 4, 6
5.) 20, 21, 29
Examples: Find the value of x. Leave your answer in simplest radical form.
6.)
7.)
16
12
x
34
10
8.)
16
x
12
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Determining Whether a Triangle is Right, Acute, or Obtuse Given Three Side
Lengths:
Right
Acute
Obtuse
Ex 9: Sides have lengths 3,
4, and 5
Ex 10: Sides have lengths
12, 6, and 11
Ex 11: Sides have lengths
14, 7 and 12
Examples: The lengths of the sides of a triangle are given. Classify the triangle as acute,
right, or obtuse.
12.) 10, 15, 20
13.) 7, 6, 4
14.) 15, 20, 25
Examples: Find the value of x. Leave your answer in simplest radical form.
15.)
16.)
22
22
x
2 3
x
36
8
2
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Algebra Review: Solve using quadratic formula
Examples:
17.) x 2  4 x  5  0
18.) 2 x 2  5  10 x
19.) 3x 2  8 x  4
20.) 4 x 2  6 x  2
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Notes #40: Chapter 7 Review
Simplify each ratio:
1.) a) BC:CD
b) m<B:m<C
B
8
C
6
c) CD: Perimeter of ABCD
A 40
2.) If x = 4, y = 6, z = 2 find each ratio:
a) x to y
6a 2 b 5
3.)
12ab7
b) (x + z) to y
c)
4.)
x y
7z
2x  y
for x  3, y  2, z  1
zx
Write and equation and solve:
4.) The ratio of the angles of a triangle is 1:3:5. Find the angles.
5.) The ratio of the angles of a pentagon is 6: 8: 9: 11: 11. Find the angles.
D
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Are the triangles similar? If so, write a similarity statement and the postulate you used:
6.)
7.)
C
6
B
10
16
8
15
A
10
E
12
D
4
8.)
9.)
E
N
12
9
D
P
O
F
15
Q
Y
M
30
18
X
24
Z
Solve for the variables:
10.)
x 1 x  4

x 3 x 8
11.)
x
12.)
y
3
12
7
5
15
22.5
x
20
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13.)
14.)
20
16
12
8
x
24
12 - x
Simplify:
15.) Similar Right Triangles: Solve for
m, n, and p in reduced radical form.
p
n
m
5
x
16.) a.) Find the geometric mean of 5 and 10
b.) Find the geometric mean of 4 and 20.
10
Are the figures sometimes, always, or never similar?
17.) two rectangles
18.) two equilateral triangles
19.) two regular hexagons
- 20 STUDY GUIDE 7
Name:________________________
Show all your work!
Date:____________Period:_______
For #1-3, ABCD is a parallelogram. Simplify each ratio:
1.) BC:CD
1.) ___________
12
A
B
2.) AD:(Perimeter of ABCD)
2.) ___________
8
120
3.) mA : mB
C
D
3.) ___________
For #4 – 6, complete each statement:
4.) If a : 3 = 7 : 4, then
4a = _____
5.) If
x 3
 , then
y 8
6.) If
x
y

, then
2 3
x2

2
y

x
4.) __________
5.) __________
6.) __________
For #7-10, solve for x:
7.)
x 15

10 25
8.)
x2 4

x3 5
7.) x = ______
8.) x = ______
9.)
x 1
x4

x2 x2
10.)
3  4x
1

1  5 x 2  3x
9.) x = ______
10.) x = ______
For #11 – 16, state whether the two polygons are always, sometimes, or never
similar.
11.) two right triangles
12.) two scalene triangles
13.) two squares
14.) two rectangles
11.) __________
12.) __________
13.) __________
14.) __________
15.) an isosceles triangle and a right
triangle
16.) two regular hexagons
15.) __________
16.) __________
- 21 For #17 - 20, refer to the diagram.
17.) Find mM '
17.) __________
A
6
18.) __________
18.) Find the scale factor of MATH to
M’A’T’H’
T
x
70
M
18
4
19.) Solve for x.
H
19.) x = ______
A'
T'
2
H'
y
M'
20.) y = ______
MATH ~ M’A’T’H’
20.) Solve for y.
For #21 – 34, complete the similarity statement and state why the triangles
are similar. If the triangles are not similar, circle not similar. (If you are
using SAS similarity or SSS similarity, be sure to check your side lengths for
a common proportion)
21.)
21.)
QRS  _____
by __________
22.)
OR
N
R
P
9
6
Q
not similar
O
S
12
16
X
12
Y
Q
M
22.)
MNO  _____
by __________
8
Z
OR
not similar
- 22 23.)
24.)
23.)
ABC  _____
by __________
F
A
12
B
21
20
8
OR
C
C
B
D
6
not similar
30
31.5
9
D
A
E
24.)
ABC  _____
by __________
OR
not similar
For #25 – 28, solve for x and y (where x and y are positive):
25.)
26.)
12
25.) x = ______
y = ______
16
6
7
x
y
9
x
12
26.) x = ______
24
27.)
28.)
27.) x = ______
15
12
x
5
24
10
9
28.) x = ______
x
Solve:
29.)
6 x 2  16 x  6
30.)
4 x 2  12 x  40  0
31.)
3 x 3  48 x
32.)
3x 2  5  9 x
29.) _______
30.) _______
31.) _______
32.) _______
- 23 For #33-34, find the geometric mean of the two numbers.
33.) 5 and 10
34.) 4 and 20
33.) ________
34.) ________
For #35-36, solve for x, y, and z. (Hint: use 3 similar, right triangles)
35.)
x
35.) x = ______
y = ______
z
y
z = ______
4
25
36.)
36.) x = ______
y
y = ______
z
4
z = ______
x
16
For # 37-38, solve for x. Leave the answer in simplified radical form
37)
38.)
37.)
x = ______
x
12
x
38.)
6
x = ______
9
For #39-40, the lengths of the sides of a triangle are given. Classify each
triangle as acute, right, or obtuse.
39.) 20, 30, 40
40.) 41, 9, 40
39.)____________
40.)__________