In this lesson you’ll learn about the concepts and the basics of Translation,
Reflection, Dilation, and Rotation. The presentation covering such content will
be done by the instructor in own handwriting, using video and with the help of
several examples with solution. This will help you understand and develop skills
for solving important geometry applications.
Translation -‘every point of the pre-image is moved the same distance in the
same direction to form the image.’ E.g. in Fig-1 above; each point (A,B,C) of the
triangle is moved to (A’,B’C’), following the rule ‘2 to the right and 1 up’ i.e.
it is moved 2 inches to the right and 1 inch up. Notice that you can
translate a pre-image to any combination of two of the four directions.
The higher level transformation may be dealt with on the coordinate plane.
The transformation in this case would be T (a, b) = (a+2, b+1).
(More text below video...)
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(Continued from above)
Translation of Axes- an equation corresponding to a set of points in the system
of coordinate axes may be simplified by taking the set of points in some other
suitable coordinate system, but all geometrical properties remain same.
One such transformation type is, in which the new axes are transformed parallel
to the original axes and origin is shifted to a new point. A transformation
of this kind is called ‘Translation
of Axes’.
Dilation- it is a type of transformation that changes the size of the image.
The size of image i.e. how much large or small size is required will determine
the 'scale factor' to be used. Suppose, we need to have 1.5 times size i.e.
using scale factor of 1.5, then the image will be 1.5 times as large as the
pre-image (Figure-2). The image will always have a prime after the letter
like- R' symbolically.
Furthermore it explains about dilation in the coordinate plane. You know from the
earlier learning about transformations in the plane that preserve distance.
There is another transformation in the plane that preserves angle measure
but not distance and this transformation is dilation. For example, in the
coordinate plane, a dilation of 2 with center at the origin will stretch
each ray by a factor of 2. If the image of A is A' then A' is a point
on and OA'
= 2 OA (Fig-3, above).
You can explore further about it, say by definition- a dilation of k is a transformation
of the plane such that:
1. The image of point O, the center of dilation, is O.
2. When k is positive and the image of P is P', then
and
the same ray and OP' = kOP.
2. When k is negative and the image of P is P', then
and
are opposite rays and OP' = -kOP.
Under dilation about a fixed point, the distance is not preserved.
In the coordinate plane, under a dilation of k with the center at the origin: P (x, y)
P’ (kx, ky) or Dk (x, y)
= (kx, ky)
The image of ABC
is
under a dilation of 1/2. A (2, 6) A'(1, 3) B (6, 4) B'(3, 2) C (4, 0) C'(2, 0)
Reflection- a reflection; also called flip, is a map that transforms an
object into its mirror image. The two common reflections are:
a horizontal reflection and a vertical reflection (Fig-4, Fig-5).
The horizontal reflection flips across and vertical reflection flips
up/down. Notice that the vertices of triangle A’,B’,C’ are corresponding
to vertices A, B, C like mirror image and the line of reflection has the
same distance from the corresponding vertex.
Line reflections in the coordinate plane: it is a transformation that
creates symmetry on the coordinate plane and to graph reflections in the
coordinate plane. E.g.
Reflection over x-axis: P(a, b) = (a, -b) Reflection over y-axis: P(a, b) = (-a,
b)
Reflection over line y = x: P(a, b) = (b, a)
You can explore further with the help of solving a case of reflection. For example:
For example: Given triangle PQR where P (–3, 1), Q (– 2, 4) and R (0, 1). Required to work
out solution to reflect the triangle across the y-axis
and sketch it.
The suggested steps are as follows:
Matrices for above reflections in the Coordinate Plane,
Reflection Theorem: Under a reflection in the origin, the image of P(a, b) is P'(-a, -b)
Given A reflection in the origin.
Prove Under a reflection in the origin, the image of P(a, b) is P'(-a, -b)
Proof Let P' be the image of P (a, b) under a reflection
in the origin, O.
=> OP = OP' and
Let B(a, 0) be the point of intersection of the x-axis and a vertical line through P and B' be the point (-a, 0). => OB = OB'and
also,
POBP'OB'
=>
POBP'OB'
(by SAS)
In particular,
PB = P'B'= |0 - b| = |b|
Since
OBP
and OB'P'
are right angles and is a vertical line.
=>
The coordinates of P' are (-a, -b).
Notice the important properties of point reflection:
• Under a point reflection, distance is preserved.
• Under a point reflection, angle measure is preserved.
• Under a point reflection, collinearity is preserved.
• Under a point reflection, midpoint is preserved.
Rotation- a rotation turns all the points in the plane around one point, called the
center of rotation i.e. the point that maps onto itself. Notice that a rotation does
not change the figures in the plane. In this context, a rotation by
180 is called
a ‘half turn’ and rotation by
90 is called a quarter turn. Generally rotations are
done counterclockwise, which is called a positive rotation and a rotation in the
clockwise direction is called a negative rotation. For example:
A rotation is a transformation
of a plane about a fixed point P through
an angle of d degrees such that:
1. For A, a point that is not the fixed point P, if the image
of A is A', then PA = PA' and m
APA'
= d.
2. The image of the center of rotation P is P.
We use RP, d as a symbol for the image under a rotation of d
degrees about point P.
Further explanation about rotation by
90
and rotation by
180:
The figure shown in (Fig-6) is a rotation by
90
and rotated around the center of rotation ‘O’.
You’ll notice that different colored lines are at right angles corresponding to each. Also
the distances from the center of rotation to corresponding vertices of the two triangles are equal.
The figure shown in (Fig-7) is a rotation by
180
and rotated around
the center of rotation ‘O’. It may be noticed that each point is in a ‘straight line’
from the center of rotation ‘O’.
Remember important statements:
• Distance is preserved under a rotation about a fixed point.
• Under a rotation about a fixed point, distance, angle measure, collinearity, and midpoint are preserved.
• Under a counterclockwise rotation of
90
about the origin, the image of P (a, b) is
P' (-b, a) and for rotation by
180
around the origin, it is P (a, b) is P' (-a, -b).
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and OA'
= 2 OA (Fig-3, above).
and
the same ray and OP' = kOP.
P’ (kx, ky) or D
ABC
is
under a dilation of 1/2.
POB
is a vertical line.
is called
a ‘half turn’ and rotation by
90
