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Geometric Similarities. Math 416. Geometric Similarities Time Frame. 1) Similarity Correspondence 2) Proportionality (SSS) (Side-side-side) 3) Proportionality (SAS) (side-angle-side) 4) Similarity Postulates 5) Deductions 6) Dimensions 7) Three Dimensions. Similarity Correspondence. - PowerPoint PPT Presentation
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Geometric Similarities
Math 416
Geometric Similarities Time Frame
1) Similarity Correspondence 2) Proportionality (SSS) (Side-side-side) 3) Proportionality (SAS) (side-angle-side) 4) Similarity Postulates 5) Deductions 6) Dimensions 7) Three Dimensions
Similarity Correspondence
Similarity – Two shapes are said to be similar if they have the same angles and their sides are proportional
Note – we see shape by angles & we see size with side length
Consider
A
C
D
B ZY
WX
Similar & Why?8
32
24 16 5
20
15 10
100
80 85
95
80
100 95
85
Proportionality (SSS)
We say the two shapes are similar because their angles are the same and their sides are proportional
We can note corresponding points A X D W B Y C Z
Angles
We note corresponding angles < ADC = < XWZ (95°) < DCB = < WZY (85°) < CBA = < ZYX (80°) < BAD = < YXW (100°)
Notes Hence we would say ADCB XWZY Hence we note corresponding angles < ADC = < XWZ < DCB = < WZY < CBA = < ZYX < BAD = < YXW
Proportionality
Next is proportionality which we will state as a fraction
AD=8 DC=16 CB=32 BA=24 XW 5 WZ 10 ZY 20 YX 15 What is the proportion (not in a
fraction)? 8/5 which is reduced to 1.6
Question #1
Identify the similar figures and state the similarity relationship, side proportion and angle equality
A
C
Z
T
C
B
BIG
SMALL
BIG
MEDMED
SMALL
Notes for Solution By observing you need to establish
the relationship. Look at angles or side lengths Important: An important trick
when comparing angles and sides is that the biggest angles is always across the biggest side, the smallest from the smallest and medium from the medium.
Solution #1
Triangle ABC ˜ TCZ AB = BC = CA TC CZ ZT < ABC = < TCZ < BCA = <CZT < CAB = <ZTC
Important Note
Make sure the middle angle letters are all different because the middle letter is the actual angle that you are looking at.
AC = CA < ACB = < BCA Both the above are the same
Question #2
K R
LT
XQ
MED
MED
SMALL
SMALL
MED
MED
With isosceles (or equilateral triangles) you may get two (or three) different answers). However, you are only required to provide
one.
Solution a for #2 The question is to identify similar figures
and state the similarity relationship, side proportion and graph equality.
QK = KT = QT RX XL RL < QKT = < RXL < KTQ = < XLR < TQK = < LRX
QKT ˜ RXL
Solution b for #2
You can also have another solution Triangle QKT is still congruent to RLX QK = KT = QT RL LX RX < QKT = < RLX < KTQ = < LXR < TQK = XRL
More Notes There are other ways of establishing
similarity in triangles At this point we will abandon reality for
simple effective but not accurate drawings of triangles… (it is not to scale).
Please complete #1 a – o For Question #3, again, state similarity
relationship, side proportion and angle equality.
Question #3
T
27
21
30
453518
Q
RCB
A
If the three sides are proportional to the corresponding three sides in the other triangle, the two will be similar.
Solution Notes
You need to check…
SMALL with SMALL
MEDIUM with MEDIUM
BIG with BIG
Solution #3
ABC ˜ QRT
MedSmall
Big
SmallMed
Big
18 = 21 = 27
30 35 45
0.6 = 0.6 = 0.6; YES SIMILAR
Proportionality SAS
We can also show similarity in triangles if we can find two set corresponding sides proportional and the contained angles equal; we can determine similarity
14°18
A
BC
ZY
X
4214°
15
35
Question #4 Show if the triangle is similar Solution… since <BAC = <XYZ = 14° 18 = 42 15 35= 6/5 6/5 BAC ˜ XYZ Notice BAC = Small, Angle, Big &
compared to Small, Angle, Big
Triangle Similarity Postulates There are three main postulates we use to
state similarity SSS all corresponding sides proportional SAS two sets of corresponding sides and
the contained angle are equal AA two angles (the third is automatically
equal since in a triangle, the interior angle must add up to 180°) are equal
Example #1
Why are the following statements true?
QPT ˜ ZXA
42°84°
84° 54°54°
AA
P T ZA
XQ
Example #2
Why are the following statements true?
KTR ˜ PMN
51°
51°
Solution: since 24/16 = 27/18
R
KT PN
M
27 1618
SAS
24
Example #3
16 12
CB
A
9
24 6
32 PT
K
Since 16 = 24 = 32
6 9 12
8/3 = 8/3 = 8/3S S S
Parallel Lines Facts: If two tranversals intersect
three parallel lines, the segments between the lines are proportional
a
b d
Therefore, a = c
b d
c
Notes
Also note that…
A
C
B BC = 1
AC 2•
•
•
Parallel and the Triangle If a parallel line to a side of a triangle
intersects the other two sides it creates two similar triangles
A
EB
CD
Therefore, ABE ˜
ACD
Question #1
3
4 x
9
Find x
3 = 9
4 x
3x = 36
x = 12
Question #2
C
x
B
40
150
50E
A
D
We note BE // CD
Thus, ABE ˜ ACD
AB = BE = AE
AC CD AD
x = 50 = AE
x+40 150 AD
Question #2 Sol’n Con’t
We only need x = 50 x+40 150 150x = 50(x+40) 150x = 50x + 2000 100x = 2000 x = 20
Proportion Ratio
Consider
1
10
1
10
Dimensions
SIDE SMALL BIG RATIO
SIDE 1D 1 10 1:10 or 1/10
AREA 2D 1 100 1:100 OR 1/100
Dimensions In general in 1D if a:b then in 2D a2 : b2
Ex. In 1D if 5:3 then in 2D? In 2D then 25:9 You can go backwards by using
square root Ex. In 2D if 36:49 then in 1D 6:7
3D or Volume
Consider
1
11 5
5
5
3D or Volume
Small Big Ratio
Side 1D 1 5 1:5 or 1/5
Volume 3D 1 125 1:125 or 1/125
3D or Volume In general in 1D if a:b then in 3D… Then in 3D a3 : b3
Ex. In 1D if 6:7 then in 3D In 3d 216:343 You can go backwards by using the
cube root Ex. In 3D if 27:8 then in 1D In 1D 3:2
Practice
Complete the following
1 D Length 2 D Area 3 D Volume
2:9 4:81 8:729
2:11 4:121 8:1331
5:3 25:9 125:27
3D Question #1
Two spheres have a volume ratio of 64:125. If the radius of the large one is 11cm, what is the radius of the small one?
Big Small r 11 x
3D Ratio 64:125
1D Ratio 4:5
Thus 4 = x
5 115x = 44
x = 8.8
3D Question #2
V=200m3
V = ?
A Base = 100m2A base = 16 m2
Question #2 Solution
Big SmallArea of Base 100 16Volume 200 x
1 Ratio 10 / 4
2 Ratio 100/16
3 Ratio 1000/64Thus 200 = 1000
x 64
1000x = 12800
x = 12.8
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