Transcript Transformations

To transform something is to change it. In geometry, there are
specific ways to describe how a figure is changed. The
transformations you will learn about include:
•Translation
•Rotation
•Reflection
•Dilation
I.
TRANSLATIONS
A Translation “slides” an object a
fixed distance in a given direction.
The original object and its translation
have the same shape and size and
they face in the same direction.
Translations are SLIDES.
Let's examine some
translations related to
coordinate geometry.
The example shows how
each vertex moves the same
distance in the same
direction.
Write the Points
• What are the coordinates
for A, B, C?
A (-4,5) B (-1,1) C (-4,-1)
• What are the coordinates
for A’, B’. C’?
A’ (2,5) B’ (5,1) C’ (2,-1)
• How are they alike?
They are similar triangles
• How are they different?
Each vertex slides 6 units to the right
Write the points
• What are the
coordinates for A, B,
C, D? A (2, 4) B (4, 4)
C (5, 2) D (2, 1)
• What are the
coordinates for A’, B’,
C’, D’? A’ (-5, 1)
B’ (-3, 1) C’ (-2, -1) D’ (-5, -2)
• How did the
transformation change
the points?
The figure slides 7 units to
the left and 3 units down
II.
ROTATIONS
A rotation is a transformation that turns a figure
about a fixed point called the center of rotation. An object
same
and its rotation are
theshape and size
, but the
figures may be turned in different directions
Rotate “About Vertex”
• Draw the first shape with the points given
• Then rotate it at the vertex (both figures will
still touch) the amount given and in the
direction given (clockwise/counterclockwise)
• Give the new points to the figure
Rotate “About Origin”
90o Rotation
(x, y)
(y, -x)
*your original point
will flip flop and
your original x
value will become
the opposite sign
180o Rotation
(x, y)
(-x, -y)
*your original point
will remain as it is,
but will become the
opposite signs
270o Rotation
(x, y)
(-y, x)
*your original point
will flip flop and
your original y
value will become
the opposite sign
These will only work if the figure is being rotated CLOCKWISE
If it’s asking for COUNTERCLOCKWISE rotation,
then 90o = 270o and 270o = 90o
Examples…
Triangle DEF has vertices
D(-4,4), E(-1,2), & F(-3,1).
What are the new
coordinates of the figure if it
is rotated 90o clockwise
around the origin?
90o … (x, y) (y, -x)
• D(-4,4) = D’(4,4)
• E(-1,2) = E’(2,1)
• F(-3,1) = F’(1,3)
D
D’
F’
E
F
E’
III.
REFLECTION
A reflectioncan be seen in water, in a mirror, in glass, or
in a shiny surface. An object and its reflection have the
same shape and size , but the figures face in opposite
directions . In a mirror, for example, right and left are
switched.
Line reflections are FLIPS!!!
The line (where a mirror may be placed) is called the
line
lineof
ofreflection
reflection. The distance from a point to the
line of reflection is the same as the distance from
the point's image to the line of reflection.
A reflection can be thought of as a "flipping" of an object over
the line of reflection.
Reflections on a Coordinate Plane
Over the x-axis
• (x, y) (x, -y)
Over the y-axis
• (x, y)
(-x, y)
*multiply the y-coordinates
by -1
(or simply take the opposite #)
*multiply the x-coordinates
by -1
(or simply take the opposite #)
What happens to points in a
Reflection?
• Name the points of the
original triangle.
A (2,-3) B (5,-4) C (2,-4)
• Name the points of the
reflected triangle.
A’ (2,3) B’ (5,4) C’ (2,4)
• What is the line of
reflection?
x-axis
• How did the points
change from the original
to the reflection?
The sign of y switches
IV.
DILATIONS
A dilation is a transformation that produces
an image that is the same shape as the
original, but is a different
different size.
size.
A dilation used to create an image larger
larger than the original is called an
enlargement. A dilation used to create an
enlargement
smaller than the original is called a
image smaller
reduction.
reduction
Dilations always involve a change in size.
Notice how
EVERY
coordinate of the
original triangle
has been
multiplied by the
scale factor (x2).
How to find a dilation
• You will multiply both the x and ycoordinates for each point by the scale
factor.
• Scale factors will be given to you.
Example…
A figure has vertices F(-1, 1),
G(1,1), H(2,-1), and I(-1,-1).
Graph the figure and the
image of the figure after a
dilation with a scale factor of
3.
F(-1, 1) = F’(-3, 3)
G(1, 1) = G’(3, 3)
H(2, -1) = H’(6, -3)
I(-1, -1) = I’(-3, -3)
How to find a scale factor
• Take the measurement of both the original
image and the dilated one and set up a ratio
• Measurement of dilation = divide #s
Measurement of original
Example…
• Through a microscope, the image of a grain
of sand with a 0.25-mm diameter appears to
have a diameter of 11.25-mm. What is the
scale factor of the dilation?
• Diameter of dilation = 11.25 =
Diameter of original 0.25
*Scale factor is 45.