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To transform something is to change it.
In geometry, there are specific ways to describe how a figure is
changed.
•Translation
•Rotation
•Reflection
•Dilation
Renaming Transformations
We use __________ _______
to help name the original shapes.
Before: PRE IMAGE
To name transformed
shapes we use the
same letters with a
_____ symbol:
After: IMAGE
A translation "slides" an object a fixed distance in a given direction.
Translations are ISOMETRIC – which means they have the same size and
the same shape!
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.
A rotation is a transformation that turns a figure about a fixed point called
the ___________________.
A Rotation is also ISOMETRIC
This rotation
is 90 degrees counterclockwise.
Clockwise
Counterclockwise
A reflection can be seen in water, in a mirror, in glass, or in a shiny surface.
A Reflection is ISOMETRIC
The line (where a mirror may be placed) is called the line of 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.
If you folded the two shapes together line of reflection the
two shapes would overlap exactly!
A dilation is NOT ISOMETRIC!
Why? A dilation used to create an image larger than the original is called an
enlargement. A dilation used to create an image smaller than the original is
called a reduction.
A vector has
to go!
and
– it tells you where
The vector <7,4> tells your x value to change by +7 and
your y value to change by + 4
Using points P,E,N,T,A, translate them along
the vector –5, –1.
P(1, 0), E(2, 2), N(4, 1), T(4, –1), and A(2, –2)
The vector indicates a translation 5 units left and
1 unit down.
(x, y)
→
(x – 5, y – 1)
P(1, 0)
E(2, 2)
N(4, 1)
T(4, –1)
A(2, –2)
→
(–4, –1)
→
(–3, 1)
→
(–1, 0)
→
(–1, –2)
→
(–3, –3)
EX 2: Translate GHJK with the vertices
G(–4, –2), H(–4, 3), J(1, 3), K(1, –2) along the vector 2, –2.
Choose the correct coordinates
for G'H'J'K'.
A. G'(–6, –4), H'(–6, 1), J'(1, 1), K'(1, –4)
B.
G'(–2, –4), H'(–2, 1), J'(3, 1), K'(3, –4)
C.
G'(–2, 0), H'(–2, 5), J'(3, 5), K'(3, 0)
D.
G'(–8, 4), H'(–8, –6), J'(2, –6), K'(2, 4)
The graph shows repeated translations that result in the
animation of the raindrop.
Describe the translation of the raindrop from position 3
to position 4 using a translation vector < a , b >.
(–1 + a, –1 + b) or (–1, –4)
–1 + a = –1
–1 + b = –4
a= 0
Answer: translation vector:
b = –3
What happens to points in a Reflection?
 Name the points of the
PRE-IMAGE:
A(,)
B (,)
C(,)
 Name the points of the
IMAGE:
A’ ( , )
B’ ( , )
C’ ( , )
 What is the line of reflection?
 How did the points change from the original to the reflection?
A reflection is always ____________from
the ___________________!
EX 1: Reflect ABCD with vertices A(1, 1), B(3, 2), C(4, –
1), and D(2, –3) across the x-axis:
Multiply the y-coordinate of each
vertex by –1.
(x, y)
→
(x, –y)
A(1, 1)
B(3, 2)
C(4, –1)
D(2, –3)
→
→
→
→
A'(1, –1)
B'(3, –2)
C'(4, 1)
D'(2, 3)
EX 2: Graph quadrilateral ABCD with vertices
A(1, 1), B(3, 2),
C(4, –1), and D(2, –3) and its reflected image
in the y-axis.
Multiply the x-coordinate of each
vertex by –1.
(x, y)
→
(–x, y)
A(1, 1)
B(3, 2)
C(4, –1)
D(2, –3)
→
→
→
→
A'(–1, 1)
B'(–3, 2)
C'(–4, –1)
D'(–2, –3)
Ex 3: Given the Pre Image Quadrilateral ABCD with vertices
A(1, 1), B(3, 2), C(4, –1), and D(2, –3).
Graph ABCD and its image
under reflection of the line y = x.
Interchange the x- and y-coordinates
of each vertex.
(x, y)
→
(y, x)
A(1, 1)
B(3, 2)
C(4, –1)
D(2, –3)
→
→
→
→
A'(1, 1)
B'(2, 3)
C'(–1, 4)
D'(–3, 2)
EX 4: Quadrilateral EFGH has vertices
E(–3, 1), F(–1, 3), G(1, 2), and H(–3, –1).
The quadrilateral was reflected, giving the image the coordinates:
E'(1, –3), F'(3, –1), G'(2, 1), H'(–1, –3)
How was the quadrilateral reflected?
It was reflected over the line y=x, since all of the
x and y values have been switched
Dilations always involve a change in size.
How did the coordiates
of the image compare
to the coordinates of
the pre image?
If k>1 the dilation is an _________________
If k<1 the dilation is a __________________
Does it make sense for k to be less than 0?
EX 1: Trapezoid EFGH has vertices
E(–8, 4), F(–4, 8),
G(8, 4) and H(–4, –8).
Graph the image of EFGH after a dilation centered at the origin with a scale
factor (k) of
Multiply the x- and y-coordinates of each vertex by the scale factor,
Ex 2: Triangle ABC has vertices A(–2, -4), B(3, 2),
C(0, 4). Graph the image of ABC after a dilation centered at the origin with
a scale factor of 1.5.
Rotation – a transformation that turns every point
of a pre-image through a specified angle and
direction about a fixed point, called the Center of
Rotation.
P is the Center of rotation!
Angle of rotation – the angle between a
pre-image point and its corresponding
image point..
The Angle of Rotation from A to A’ is 90°
Counterclockwise!
EX 1: For the diagram to the right, which
description best identifies the rotation of
triangle ABC around point Q?
A. 20° clockwise
B. 20° counterclockwise
C. 90° clockwise
D. 90° counterclockwise
Thankfully, we really only care about rotating three ways…
EX 2: Hexagon DGJTSR is shown below. What is the image of point T after a
90 counterclockwise rotation about the origin?
A (5, –3)
B (–5, –3)
C (–3, 5)
D (3, –5)
EX 3:
1.
What is the image of point Q after a 90° counterclockwise
rotation about the origin?
(–5, 4)
2. What is the image of point Q after a 180° counterclockwise
rotation about the origin?
(-4, -5)
3. What is the image of point Q after a 270° counterclockwise
rotation about the origin?
(5, -4)
Rotational Symmetry:
A figure in a plane has rotational symmetry if the figure can be mapped onto
itself by a rotation of 180⁰ or less.
The figure above has rotational symmetry because it maps onto itself by a
rotation of 90⁰.
When a figure can be rotated less than 360° and the image and pre-image
are indistinguishable (the same)
Symmetry
Rotational: 120°
Lines of :
Symmetry?
3
360/3 = 120
90°
60°
45°
4
6
8
360/4 = 90
360/6 = 60
360/8 = 45
Does the following shape have
symmetry?
Lines of Symmetry?
_______________
Rotational Symmetry?
_______________
Day 5 Composites!
Composite transformations mean:
More than one transformation is
happening!
Original
Move 1
Move 2
EX 1: Quadrilateral BGTS has vertices
B (–3, 4), G (–1, 3), T(–1 , 1), and S (–4, 2).
Graph BGTS and its image after a translation along 5, 0 and a reflection in
the x-axis.
Step 1
translation along 5, 0
(x, y)
→
(x + 5, y)
B(–3, 4)
G(–1, 3)
S(–4, 2)
T(–1, 1)
→
B'(2, 4)
G'(4, 3)
S'(1, 2)
T'(4, 1)
→
→
→
Step 2
reflection in the x-axis
(x, y)
→
(x, –y)
B'(2, 4)
G'(4, 3)
S'(1, 2)
T'(4, 1)
→
B''(2, –4)
G''(4, –3)
S''(1, –2)
T''(4, –1)
→
→
→
Ex 2:
Quadrilateral RSTU has vertices
R(1, –1), S 4, –2), T (3, –4), and U (1, –3).
Graph RSTU and its image after a translation along –4, 1 and a reflection
in the x-axis. Which point is located at (–3, 0)?
A. R'
B. S'
C. T'
D. U'
EX 3:
ΔTUV has vertices T(2, –1), U(5, –2), and V(3, –4). Graph ΔTUV and its
image after a translation along –1 , 5 and a rotation 180° about the
origin.
Step 1
translation along –1 , 5
(x, y)
→
(x + (–1), y + 5)
T(2, –1)
U(5, –2)
V(3, –4)
→
T'(1, 4)
U'(4, 3)
V'(2, 1)
→
→
Step 2
rotation 180 about the origin
(x, y)
→
(–x, –y)
T'(1, 4)
U'(4, 3)
V'(2, 1)
→
T''(–1, –4)
U''(–4, –3)
V''(–2, –1)
→
→
EX 4: ΔJKL has vertices J(2, 3), K(5, 2), and L(3, 0). Graph ΔTUV and its
image after a translation along 3, 1 and a rotation 180° about the origin.
What are the new coordinates of L''?
A.
(–3, –1)
B.
(–6, –1)
C.
(1, 6)
D.
(–1, –6)