Download similarity has a lot to do with proportionality

CST4_Lesson23_SimTri
March 17, 2017
Triangle Geometry
Similar Triangles
similarity has a lot to do with proportionality
the mathematical understanding of SIMILAR is different than our "normal" understanding of the word
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CST4_Lesson23_SimTri
March 17, 2017
Two figures are considered SIMILAR if:
....they have the same shape
....the corresponding angles are congruent
....the corresponding sides are proportional
All three of the above conditions must be met When things grow proportionally, we say that they grow in all dimensions by the same factor
if something gets three times as wide, it will also get three times as tall and three times as deep 2
CST4_Lesson23_SimTri
March 17, 2017
If two parallelograms are proportional, then the corresponding sides are larger (or smaller) by the same factor.
scale factor = 5
2
scale factor = 2
5
8.5 cm
3.4 cm
3 cm
these are proportional
7.5 cm
increasing the size, then k > 1 decreasing the size, then k < 1
A proportion is made of two equal ratios (fractions)
from the previous page: 3.4 3
=
8.5 7.5
the measure of a side from "the first" figure
the measure of its corresponding side in the second figure
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CST4_Lesson23_SimTri
March 17, 2017
a c
=
b d
a proportion
after we cross multiply, these will be equal
ad=bc
ex 3.4 3
=
8.5 7.5
Two figures are considered SIMILAR if
....they have the same shape
this you can verify by looking...is it a triangle, trapezoid, square etc...
....the corresponding angles are congruent
this you can verify mathematically ...the measures are either given or can be calculated or deduced
....the corresponding sides are proportional
this you can verify mathematically ...set up a proportion 5
8
9
14.4
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CST4_Lesson23_SimTri
March 17, 2017
Remember!
Once a triangle has been proven to be similar to another triangle, then you know two things:
corresponding angles are congruent
corresponding sides are proportional
Like proving congruence...we also have minimum conditions to prove SIMILARITY
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CST4_Lesson23_SimTri
March 17, 2017
3 ways to prove that triangles are similar:
1. AA ­­ angle­angle
2. SSSprop ­­ side­side­side (prop)
3. SASprop ­­ side­angle­side (prop)
1. Proving triangles are similar by AA
Angle­Angle
statement
Justification 6
CST4_Lesson23_SimTri
March 17, 2017
Now that the 2 triangles are similar by AA
ED x k = AB
EF x k = BC
DF x k = AC
where k is the ratio of similarity.
sides in one triangle are k times the corresponding sides in the other triangle
k can be greater than 1 or less than 1
What are possible values for k ?
Example of a question:
D
Determine the length of segment AB
A
C
55
94.4 cm
20 cm
B
55
E
statement
59 cm
F
justification 7
CST4_Lesson23_SimTri
March 17, 2017
Example of a question:
D
A
Determine the length of segment AB
mc 02
C
B
55
94.4 cm
55
E
59 cm
F
justification statement
2. Proving triangles are similar by SSSprop
side­side­side (prop)
7
10.5
6
9
9
13.5
Statement
Justification 8
CST4_Lesson23_SimTri
March 17, 2017
Now that the 2 triangles are similar by SSSprop
7
10.5
ED x k = AB
EF x k = BC
DF x k = AC
6
9
9
13.5
Where k is the ratio of similarity.
sides is one triangle are k times the corresponding sides in the other triangle
corresponding angles are congruent
What are possible values for k ?
Example of a question:
Prove that C F
A
D
91
96
174.4
60
E
F
B
C
statement
justification 9
CST4_Lesson23_SimTri
March 17, 2017
3. Proving triangles are similar by SASprop
side­angle­side (prop)
6
10
8.4
14
justification statement Now that the 2 triangles are similar by SASprop
A
ED x k = AB
EF x k = AC
DF x k = BC
D
91
96
174.4
60
E
F
B
C
Where k is the ratio of similarity.
sides in one triangle are k times the corresponding sides in the other triangle
What are possible values for k ?
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CST4_Lesson23_SimTri
March 17, 2017
Example of a question:
Can you determine the length of EF?
A
F
14
11
8
D
B
5
E
C
justification statement Example of a question:
What is the length of EF?
A
F
14
11
8
D
E
19.25
B
5
C
statement justification 11
CST4_Lesson23_SimTri
Dealing with OVERLAPPING TRIANGLES
March 17, 2017
Many problems involving similar triangles have one triangle ON TOP OF (overlapping) another triangle. Here we have ∆BDE and ∆BAC
In this case line segments DE and AC are shown to be parallel­­note the arrows on the line segments.
so, we know that <BDE is congruent to <DAC (by corresponding angles). <B is shared by both triangles
so the two triangles are similar by AA
Parallel Line to a Triangle’s Side: Any line parallel to a triangle’s side determines two similar triangles.
∆BXY ∆BAC
angles are congruent sides are proportional
Corollary: Any two similar triangle determine parallel lines.
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CST4_Lesson23_SimTri
March 17, 2017
Segment Joining the Midpoints of Two Sides in a Triangle: Any segment joining the midpoints of two sides in a triangle is parallel to the third side and is half the measure of this third side.
Thales’ Theorem: Two intersecting transversal lines intersected by parallel lines are separated into corresponding segments of proportional length.
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CST4_Lesson23_SimTri
March 17, 2017
A
D
B
E
C
More OVERLAPPING TRIANGLES
A
A
D
D
B
B
C
E
C
E
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CST4_Lesson23_SimTri
March 17, 2017
AB CD EF GH
Lines Does not have to be a triangle
G
E
C
A
O
B
D
F
H
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