Transformation Geometry
For Students
Transformation Geometry
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Transformation geometry is the study of
figures that move under certain
conditions.
In other words it is the study of moving
a shape from one position to another
Transformations
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There are two categories of Transformation
Geometry, these are:
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Isometries
Similarities
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Isometries, the shape &size of the object do not
change
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Similarities the shape of the object is retained but
the size changes
OBJECT before transformation
IMAGE after transformation
Isometries
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This category is broken into four
different transformations, these are:
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Translation
Axial symmetry
Central symmetry
Rotation
Similarities
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This category is broken up into two
different categories, these are:
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Enlargements
Reductions
Translations
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Under a translation all points of an
image are moved the same distance in
the same direction
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Relate translation to an object on a conveyor
belt. On a conveyor belt all objects move the
same direction, the same distance apart and in
the same manner.
Translation
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Every translation has a translation
vector, this tells us the distance and the
direction in which the object is moved
Axial symmetry
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Under axial symmetry, any point and its
image are mirrored the same distance
away from a plane as it is to the plane.
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axial symmetry can be related to a
mirror image of an object, we see
everything in reverse. Also a reflection of
the object is the same distance form the
mirror as the image appears to be from
the mirror
Axis of Reflection
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Axis of Reflection—this is the line of
axis that the object is flipped over to
become the image. The rule is that the
object is projected perpendicular
through this line
Rules of Axial Symmetry
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With axial symmetry we see that:
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An object and its image are congruent (equidistant)
The position of a point on the axis of reflection stays
in the same position
The axis of reflection, being the locus of all points
equidistant from a point and its image, is the
perpendicular bisector of a line segment joining a
point and its image.
A point and its image are equidistant from the axis of
reflection
Central Symmetry
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Central symmetry is the reflection of an
object through a point and on the same
distance at the other side.
Centre of Symmetry
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This is where the points of the object are projected
through and out equidistant on the other side.
In central symmetry the image is always upside down
and back to front compared to the object.
Rotation
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Under a rotation, a figure is:
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Turned about a fixed point
Through a certain angle
The fixed point in which the object is
rotated is called the centre of rotation
The angle in which it is turned about is
called the angle of rotation.
Enlargements
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Under an enlargement an object is projected
onto a similar object and a parallel line.
Centre of enlargement—is the point from
where the enlargement measuring from.
Scale factor—The length of the enlargement,
this is given in a ratio to the length of the
original.
Reduction
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Reduction is the exact same as
enlargement except the opposite way
around!!!
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centre of reduction
Axial symmetry—Rotation
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A combination of two axial symmetries
is a rotation.
Axial symmetry—Central
symmetry
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A composition of two axial symmetries
in a perpendicular axes is a central
symmetry
Central symmetry--Translation
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A composition of two central
symmetries is a translation
Theorems in Transformation
Geometry
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Parallel projection conserves equality and proportion of length.
Translation maps a line onto a parallel line.
Central symmetry maps a line onto a parallel line.
The composition of two central symmetries (point reflections) is
equivalent to a translation.
The composition of two axial symmetries (line reflections), in
intersecting axes, is equivalent to a rotation.
The composition of two axial symmetries in perpendicular axes is a
central symmetry in their point of intersection.
The composition of two axial symmetries in parallel axes is a
translation.
An enlargement maps a line onto a parallel line.
Axial symmetry, central symmetry, translations and their composite
mappings are isometries (conserving measure of length, angle and
area).
The composition of translations is both commutative and associative.
The composition of reflections is neither commutative nor associative.
TERMINOLOGY
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ANTICLOCKWISE:
Rotating to the left as you look at the page.
AXIAL SYMMETRY:
To flip an object over a line.
BISECT:
To bisect something means to cut it into two equal halves.
CENTRAL SYMMETRY:
Reflection of an object through a point.
CLOCKWISE:
Rotating to the right as you look at the page.
CONGRUENT:
Congruent means equal in all respects.
ENLARGEMENT:
To make an object bigger.
INTERSECTION:
Where the two lines cross/meet.
IMAGE:
Outcome of the object after it ahs been transformed
ISOMETRY
Object moves under a transformation but the size doesn’t
LINE SEGMENT:
A line segment begins and ends at particular points
OBJECT:
Original figure before it is moved under its transformation
PARALLEL LINES:
Lines that never meet. They run side by side
PERPENDICULAR LINES:
Lines that meet at a right angle (90º).
PLANE:
A plane is a flat surface that extends in all directions.
REDUCTION:
To make an object smaller
RIGHT ANGLE:
A right angle measures 90°.
ROTATION:
To turn an object around a fixed point.
SIMILARITY:
The size of the object changes but the shape stays the same.
THEOREM:
A theorem is a proven mathematical statement.
TRANSLATION:
To move an object means by sliding it without rotating it.
TRANSFORMATION GEOMETRY: The study of figures that move under certain conditions.