Transcript 2.8 – Proportions & Similar Figures

Proportions &
Similar Figures
VOCABULARY
1. Similar Figures - Figures that are the same shape,
but not the same size. Corresponding angles are
equal, but the corresponding sides are proportional.
2. Scale – the ratio of any length on a drawing to the
actual length
Symbols
~
~
=
Means is similar to
Means is congruent to
Similar Figures
A
3 cm
2 cm
B
D
2 cm
3 cm
W
9 cm
6 cm
Z
6 cm
C
X
Corresponding sides:
AB corresponds to WX.
BC corresponds to XY.
CD corresponds to YZ.
AD corresponds to WZ.
9 cm
Y
Similar Figures
A
3 cm
2 cm
B
D
2 cm
3 cm
W
9 cm
6 cm
Z
6 cm
C
Corresponding angles:
A corresponds to
B corresponds to
W.
X.
C corresponds to
D corresponds to
Y.
Z.
X
9 cm
Y
Similar Figures
A
3 cm
2 cm
B
D
2 cm
3 cm
W
9 cm
6 cm
Z
6 cm
C
X
9 cm
In the rectangles above, one proportion is
AB
AD
2
3
=
, or
= .
WX WZ
6
9
Y
If you cannot use corresponding side lengths to
write a proportion, or if corresponding angles are
not congruent, then the figures are not similar.
Missing Measures in Similar Figures
The two triangles are similar. Find the
missing length y and the measure of D.
100
111
____
___
200 = y
Write a proportion using
corresponding side lengths.
200 • 111 = 100 • y Cross multiply.
22,200 = 100y
Simplify & Divide by 100.
100 = 100
Y = 222mm
The two triangles are similar. Find
the measure of angle D.
Angle D is congruent to angle C.
If angle C = 70°, then angle D = 70° .
Try This
The two triangles are similar. Find the
missing length y and the measure of B.
A 60 m
65°
50 m
45° 52 m
B
120 m
100 m
y
50
52
____
___
=
100
y
5,200 = 50y
5,200 = 50y
_____
___
50
50
104 m = y
Write a proportion using
corresponding side lengths.
Divide both sides by 50 to
undo the multiplication.
Try This
The two triangles are similar. Find the
missing length y and the measure of B.
A 60 m
65°
50 m
45° 52 m
B
120 m
100 m
y
Angle B is congruent to angle A.
If angle A = 65°, then angle B = 65°
- These figures are congruent
.
- They have the same shape
and
the same size.
- These figures are similar
.
- They have the same shape
, their
angles are congruentand their
sides are proportional
.
Since similar figures are proportional, we
can use proportions to solve for missing
measures.
Matching corresponding sides just means you
are going to match the long side of one figure
with the long side on the other figure, and so on.
A
B
C
D
E
F
Match the corresponding sides on the
figures below. Large
Small
Side A
Side B
Side C
A
D
___
E
___
F
___
B
D
C
E
F
Use proportions to solve for side B.
___
5
4
Short
Long
7
B
5
___
B
6.88
8.6
F
6.88
4
Use proportions to solve for side F.
Medium
Short
7
8.6
5
___
7
F
5
___
4
F
5.6
6.88
4
Using Proportions with Similar Figures
This reduction is similar to a picture
that Katie painted. The height of the
actual painting is 54 centimeters. What
is the width of the actual painting?
Actual
Reduced
2
54
3
w
Using Proportions with Similar Figures
Actual
Reduced
2
54
3
w
2 cm
3 cm
_____
=
Write a proportion.
w
cm
54 cm
54 • 3 = 2 • w Cross multiply.
162 = 2w
81 = w
simplify
Divide both sides by 2 to
undo the multiplication.
Try these 5 problems.
These two triangles are similar.
1. Find the missing length x. 30 in.
2. Find the measure of
J.
36.9°
3. Find the missing length y. 4 in.
4. Find the measure of
P.
90°
5. Susan is making a wood deck from plans for
an 8 ft by 10 ft deck. However, she is going to
increase its size proportionally. If the length is
12 ft
to be 15 ft, what will the width be?
For Polygons to be Similar
corresponding angles must be
congruent,
and
corresponding sides must be
proportional
(in other words the sides must
have lengths that form
equivalent ratios)
Just as we solved for variables in
earlier proportions, we can solve
for variables to find unknown sides
in similar figures.
Set up the corresponding sides as a
proportion and then solve for x.
x
5
12
10
10x = 60
x = 6
Ratios
x/12
and
5/10
Determine if the two triangles are similar.
The two windows
below are similar.
Find the unknown
width of the larger
window.
These two buildings are
similar. Find the height
of the large building.
Find the missing sides
lengths, a and b.
First, write a ratio you know
8 .
6.4
Next, write a proportion with a
side length you need to find.
8 = 7 .
6.4
a
Last, Cross multiply and solve!
8a = 44.8
a = 5.6
Find length
b on your
own
The sun’s rays strike the building and
the girl at the same angle, forming
two similar triangles. How tall is the
building if the girl is 5ft?
girl’s shadow = building’s shadow
girl’s height
building’s height
3 = 15
5
x
3x = 75
x = 25
The building
is 25 feet
tall.
x
-5ft-3ft-
-------15ft--------