Transformations—Part II —Size Transformations ( Dilation )
S68
Imagine a figure centered on the origin of the Cartesian plane, w hen a sudden inflation of the plane
occurs, so that every point is pushed out tw ice as far from the origin as w hen it started.
Innocent figure,
sitting in plane
Suddenly, w oosh! Every point
is pushed out tw ice as far from
the origin (as before).
The figure is blow n up
to “ tw ice” its original size!
But even figures not centered on the origin get “ blow n up” into a new figure, same shape,
but “ tw ice as large” :
R
R
C
Figure aw ay from
center of dilation...
Is moved
Tw ice as far aw ay...
So ev ery dist ance is doubled, and t he
resulting figure is “ tw ice the size” , but still
the same shape.
To see w hy it must be the same shape, consider the follow ing dilation by a FACTOR of 3:
C
C
R
A
2R
A
2
N
E
1
B
D
N
Look at triangle CAB
After the dilation,
the new points are C A N BN.
B
Why are triangles C A N BN and CAB similar?
E
N
What angle has the same measure as
p CAB (2) ?
N
D
(Why?)
What can be said about segments AB and A N BN ?
NN
Could the same arguments be made regarding CBD and CB D ?
NN
NN
NN
(And thus BD & B D ... ?)
Finally, if AB | | A B and BD | | B D ,
then the angle formed by AB and BD and the angle formed by A B and B D must be....
NN
NN
The conclusion: If a figure is transformed by a dilation, every angle w ithin the figure is preserved, so
the new figure has the same shape as the original.
Size Transformations ( Dilation ) —continued
S68
Notice that in the first transformation, every point w as moved to a new location twice as far from
the center of the dilation. Thus all distances w ere doubled.
A side of the new figure is 200% as large as the original.
It is, how ever, only 100% larger than the original.
In the second transformation, every point w as moved to a new location 3 times as far from the
center of the dilation. Thus all distances w ere tripled.
A side (eg A B ) of the new figure is 300% A S LA RGE as (3 times as large as) the original (AB).
(It is 200% larger than (2 times larger than) the original.)
NN
Exercises:
1.
A figure is dilated so that its points move 1.5 times as far from the center of dilation.
10.5m
a. Find the dimensions of the parts of the smaller figure
w hich correspond to those given in the larger figure.
9 m
8.7 m
If the area of the smaller figure is 43 m 2 ,
w hat must be the area of the smaller figure?
b.
11.4m
2.
a. Are the tw o given figures similar?
b. What are the lengths of the 3 rd sides?
c. If the area of the left triangle is 27 cm2 ,
then the area of the other triangle
must be ...
9
12
d. Approximately w here w ould be the
center of the dilation ? (Show it.)
3.
6
a. What is the scale factor in problem #2?
b. Each edge of the figure on the right is
figure on the left.
times as large as the corresponding edge of the
c. Each edge of the figure on the left is
figure on the right.
times as large as the corresponding edge of the
d. Each edge of the figure on the right is
the figure on the left.
times larger than the corresponding edge of
e. Each edge of the figure on the left is
the figure on the right.
4.
8
times smaller than the corresponding edge of
Find the value of x & y using pairs of similar triangles below :
B
x
D
B
6
y
²E
10
12
A
x
ú
13
F
E
D
12
12
y
7.5
A
C
25
Here it is given that AB | | AC
and AB and DC intersect at E
x= 1 5 , y= 2 0
F
B
x
Here pA & pD are both right,
And AD and BE intersect at F.
x = 1 8 , y = 1 9 .5
C
5
A
pC is right and CF z AB
y = 1 3 , x = 1 2 @ 5 /1 3