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Chapter 7Similarity and Proportion
• Express a ratio in simplest form.
• State and apply the properties of similar polygons.
• Use the theorems about similar triangles.
7.1 Ratio and Proportion
Objectives
• Express a ratio in simplest form
• Solve for an unknown in a proportion
Ratio
• A comparison between numbers
12
35
69
ab
5 : 7
5 : 7 : s : 5 : t
Ratio
• Always reduce ratios to the simplest form
Proportion
• An equation containing ratios
5
8
a
b
3
5
a
b
3
5 15
a
3 4
5 2
a d
b c
Solving a Proportion
3
5 15
a First, cross-multiply
5 45a Next, divide by 5
9a
White Board Practice
• ABCD is a parallelogram. Find the value of each ratio.
A
D C
B
10
6
White Board Practice
• AB : BC
A
D C
B
10
6
White Board Practice
• 5 : 3
A
D C
B
10
6
White Board Practice
• BC : AD
A
D C
B
10
6
White Board Practice
• 1 : 1
A
D C
B
10
6
White Board Practice
• m A : m C
A
D C
B
10
6
White Board Practice
• 1 : 1
A
D C
B
10
6
White Board Practice
• AB : perimeter of ABCD
A
D C
B
10
6
White Board Practice
• 5 : 16
A
D C
B
10
6
White Board Practice
• x = 2 and y = 3. Write each ratio in simplest form.
x to y
White Board Practice
• x = 2 and y = 3. Write each ratio in simplest form.
2 to 3
White Board Practice
• x = 2 and y = 3. Write each ratio in simplest form.
6x2 to 12xy
White Board Practice
• x = 2 and y = 3. Write each ratio in simplest form.
1 to 3
White Board Practice
• x = 2 and y = 3. Write each ratio in simplest form.
y – x
x
White Board Practice
• x = 2 and y = 3. Write each ratio in simplest form.
1
2
7.2 Properties of Proportions
Objectives
• Express a given proportion in an equivalent form.
Means and Extremes
• The extremes of a proportion are the first and last terms• The means of a proportion are the middle terms
=a
b
c
da : b = c : d
Properties of Proportion
a c
b d is equivalent to
a b
c dad bc b d
a c
a b c d
b d
1.
4.
3. 2.
That just means that you can rewrite
5
2
x
yyx 52
2
25
y
yx
As any of these
1.
4.
3. 2. 25
yx
2
5
y
x
Another Property
...If ... then
...
a c e a a c e
b d f b b d f
White Board Practice
• If , then 2x = _______2
4
7
x
White Board Practice
• If , then 2x = 282
4
7
x
White Board Practice
• If 2x = 3y, then 3
2
White Board Practice
• If 2x = 3y, then x
y
3
2
White Board Practice
• If , then2
4
7
x
7
7x
White Board Practice
• If , then2
4
7
x
2
6
7
7
x
White Board Practice
• If , then2
2
3
yx 3
3x
White Board Practice
• If , then2
2
3
yx23
3 yx
7.3 Similar Polygons
Objectives
• State and apply the properties of similar polygons.
Similar Polygons
• Same shape
• Not the same size Why?
• Not the same size Why?
Because then they would be congruent !
Similar Polygons (~)• All corresponding angles congruent
A A’
B B’
C C’A
B
C
A’
B’ C’
Similar Polygons (~)
• All corresponding sides in proportion
AB = BC = CA
A’B’ B’C’ C’A’A
B C
A’
B’ C’
The Scale Factor
The reduced ratio between any pair of corresponding sides or the perimeters.
12:3
12
3
Finding Missing Pieces
You have to know the scale factor first to find missing pieces.
12
3
y10
y
10
3
12
White Board Practice
• Quadrilateral ABCD ~ Quadrilateral A’B’C’D’. Find their scale factor
A
D C
B
A’
D’ C’
B’
50
y
30 20 12
x z
30
White Board Practice
• 5:3
A
D C
B
A’
D’ C’
B’
50
y
30 20 12
x z
30
White Board Practice
• Quadrilateral ABCD ~ Quadrilateral A’B’C’D’. Find the values of x, y, and z
A
D C
B
A’
D’ C’
B’
50
y
30 20 12
x z
30
White Board Practice
• x = 18
• y = 20
• z = 13.2A
D C
B
A’
D’ C’
B’
50
y
30 20 12
x z
30
White Board Practice
• Quadrilateral ABCD ~ Quadrilateral A’B’C’D’. Find the ratio of the perimeters
A
D C
B
A’
D’ C’
B’
50
y
30 20 12
x z
30
White Board Practice
• 5:3
A
D C
B
A’
D’ C’
B’
50
y
30 20 12
x z
30
7.4 A Postulate for Similar Triangles
Objectives
• Learn to prove triangles are similar.
AA Simliarity Postulate(AA~ Post)
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
A
B C
D
EF
Remote Time
• T – Similar Triangles
• F – Not Similar
T – Similar TrianglesF – Not Similar
T – Similar TrianglesF – Not Similar
T – Similar TrianglesF – Not Similar
T – Similar TrianglesF – Not Similar
7-5: Theorems for Similar Triangles
Objectives
• More ways to prove triangles are similar.
SAS Similarity Theorem (SAS~)
If an angle of a triangle is congruent to an angle of another triangle and the sides including those angles are proportional, then the triangles are similar. A
B C
D
EF
SSS Similarity Theorem (SSS~)
If the three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar.
A
B C
D
EF
Homework Set 7.5
• 7-5 #1-19 odd
• WS PM 40
7-6: Proportional Lengths
Objectives
• Apply the Triangle Proportionality Theorem and its corollary
• State and apply the Triangle Angle-bisector Theorem
Divided ProportionallyIf points are placed on segments AB and CD so that , then we say that these
segments are divided proportionally.AX CY
XB YD
A
X
B
C
Y
D
See It!
Theorem 7-3If a line parallel to one side of a triangle intersects the
other two sides, it divides them proportionally.
X
Y
Z
A B
See It!
CorollaryIf three parallel lines intersect two transversals,
they they divide the transversal proportionally.
R
S
T
W
X
Y
See It!
Theorem 7-4If a ray bisects an angle of a triangle, then it divides the
opposite side into segments proportional to the other two sides.
X
Y
Z
W
See It!
Construction 12Given a segment, divide the segment into a given number of congruent segments.
Given:
Construct:
Steps:
AB 1a segment of length
3CD AB
Construction 13Given three segments, construct a fourth segment so that the four segments are proportional.
Given:
Construct:
Steps:
, , AB CD EF AB EF so that
CD XYXY
Homework Set 7.6
• WS PM 41
• WS Constructions 12 and 13
• 7-6 #1-23 odd
• Quiz next class day
