Geometric transformations are
operations that change the
position, shape, or size of a
figure1.
1. Translation
2. Reflection
3. Rotation
4. Dilation
Transformations can be combined to
create more complex
transformations2. Transformations can
also be expressed algebraically using
graph functions2.
Transforma
Function
tion
Translation Slides or moves
the pre-image
Rotation
Rotates or turns
the pre-image
around an axis
Result
No change in size or
shape; Changes only
the direction of the
shape.
No change in size
or shape
Transforma
Function
tion
Reflection Flips the pre-image
Dilation
and produces the
mirror-image
Stretches or shrinks
the pre-image
Result
No change in size or
shape or orientation
Expands or contracts
the shape
Consider a function f(x). On a coordinate
grid, we use the x-axis and y-axis to
measure the movement. Transformations
can be represented algebraically and
graphically. We can use the formula of
transformations in graphical functions to
obtain the graph just by transforming the
basic or the parent function, and thereby
move the graph around, rather than
tabulating the coordinate values.
Transformations help us visualize and
Translation of a 2-d shape causes sliding
of that shape. We need to find the
positions of A′, B′, and C′ comparing its
position with respect to the points A, B,
and C. We find that A′, B′, and C′ are:
•8 units to the left of A, B, and C
respectively.
•3 units below A, B, and C respectively.
Translation
This translation can algebraically be
translated as 8 units left and 3 units
down. i.e. (x,y) → (x-8, y-3)
Example: (x,y) → (x-8, y-3)
Example: (x,y) → (x-8, y-3)
Reflection. Every point (p, q) is reflected
onto an image point (q, p). If point A is 3
units away from the line of reflection to
the right of the line, then point A' will be 3
units away from the line of reflection to
the left of the line. Thus the line of
reflection acts as a perpendicular
bisector between the corresponding
points of the image and the pre-image.
Example: (x,y) → (-x, 2-y)
(-2,4) →(2,-2), (-3,1) → (3,1) and (0,1 ) →
(0,1)
Transformation of Rotation
The general rule of transformation of
rotation about the origin is as follows.
To rotate 90º: (x,y) → (-y, x)
To rotate 180º (x,y) → (-x,-y)
To rotate 270º (x,y) → (y, -x)
Example: (x,y) → (-x, 2-y)
(-2,4) →(2,-2), (-3,1) → (3,1) and (0,1 ) →
(0,1)
Dilation transformation rules
1.If a point (x, y) is dilated by a factor of c
about the origin, its image is the point (cx,
cy).
2.More generally, if the point (x, y) is dilated
by a factor of c about the point (a, b), its
image is the point (c (x - a) + a, c (y - b) +
b).
3.The center of dilation is a fixed point, and
the image of any other point is determined
by the scale factor.
Identify the word that is being ask on the
question.
1. A turn is also called ________________.
2. Translation, reflection and rotation are also
known as ________________.
3. What is the rule of the coordinate given the
illustration?
Identify whether the transformation tells
that it is translation, rotation, dilation or
reflection.
1. (x, y) → (x, -y)
2. (x, y) → (x, 4y)
3. (x, y) → (5x, 5y)
4. (x, y) → (x+2, y-3)
5. (x, y) → (-x, y)
P∈l
Q∈l
P∈l
m∈l
l = PQ
l joins P and Q
l = mn = nm
l joins two lines m and n.
l = mP = Pm
l joins line m and point P.
l = PQ
l joins P and Q
l = mn = nm
l joins two lines m and n.
l = mP = Pm
l joins line m and point P.
𝛼
Note:
lm is a plane.
l ∙ m is a point.
l ∙ 𝛼 is a common point of a line and a plane.
l ∙ 𝛼 is a common point of a line and a plane.
𝛼 ∙ 𝛽 is a common point of two planes.
Triangle:
Vertices: P, Q, and R.
Sides: PQ, QR, and PR.
A pencil is a family of geometric objects with a
common property, for example the set of lines
that pass through a given point in a plane,
pencil of lines, pencil of circles, pencil of
spheres, pencil of conics.
Range-set of all points in a line. Section of the
pencil or the center of perspectivity.
What happen to
lines and projective
plane?
Will you give a real
life representation of
projectives?
Pencil-set of all lines that lie in a plane and
pairs through the points. Pencil projects the
range.
Range-set of all points in a line. Section of the pencil.
Axioms in Projective Geometry
A1: Two distinct points P and Q lie on
exactly one line.
A2: Any two lines meet in at least one
point.
A3: There exist three collinear points.
A4: Every line contains atleast three
points
O
N
P
M
Q
R
Complete quadrilateral is a system
of four lines, no three of which pass
through the same point, and the six
points of intersection of these lines.
Definition:
If four points in a plane are
joined in pairs by six distinct lines,
they are called the vertices of a
complete quadrangle, and the lines
are six sides.
Quadrangle:
PQRS.
Sides: PQ, QR, RS, SP, PR and QS.
Vertices: P, Q, R, and S.
Opposite Sides
Diagonal Point
PR and QS
PS and QR
PQ and SR
B
A
C
Definition:
If four lines in a plane meet by
six distinct points, they are called
the sides of a complete quadrangle,
and the points are six vertices.
A complete quadrilateral are four
points making up a quadrilateral
are joined pairwise by six distinct
lines.
pqrs
Complete Quadrilateral:
Vertices: p∙s p∙r p∙ 𝑞 r∙ s q∙ r q∙ s
s
Sides: p q r
Opposite Vertices
p∙s and q∙ r
p∙ q and r∙ s
p∙ r and q∙ s
Diagonal Line
a
c
b
1. Name the quadrilateral.
2. Name the opposite sides.
3. Name the opposite vertices.