2.
Transformation geometry is the geometry of moving
points and shapes.
• The type of transformation dealt with in this
module is:
• Translations of p units horizontally and q
units vertically.
• A translation is a horizontal or vertical slide.
• The object translated does not change its
shape or size, that is the object and the
image are congruent.
3.
TRANSLATION OF POINTS
• Let us first revise the
plotting of points on the
cartesian plane.
• Plot the following points on
the grid provided.
• A(2;4), B(-3;6),C(-5;-6),
• D(6;-4)
• Now translate each point 2
units to the right and 1 unit
downward.
4.
EXAMPLE ONE
• Consider ∆ABC in the figure
alongside.
• ∆ABC has been translated 10
units to the left to form the
image ∆A’B’C’.
• You will notice that the three
vertices of the ∆ABC has moved
10 units to the left.
• A has moved 10 units left to
form A’.
• B has moved 10 units left to
form B’.
• C has moved 10 units left to
form C’.
• ∆ABC is congruent to ∆A’B’C’.
They are identical in size and
shape.
5.
EXAMPLE TWO
• Consider ∆ABC in the figure below.
• ∆ABC has been translated 9units
downwards to form the image
∆A’B’C’.
• You will notice that the three
vertices of the ∆ABC has moved
9units downward.
• A has moved 9 units downward to
form A’.
• B has moved 9 units downward to
form B’.
• C has moved 9 units downward to
form C’.
• ∆ABC is congruent to ∆A’B’C’. They
are identical in size and shape.
6.
EXAMPLE THREE
• In this example, ABC has first
translated 11 units to the left and
then 9 units downwards.
• Notice that the three vertices have
moved 11 units to the left and then
9 units downwards.
• A has moved 11 units to the left
and then 9 units downward to
form A‘
• B has moved 11 units to the left
and then 9 units downward to
form B‘
• C has moved 11 units to the left
and then 9 units downward to
form C‘
• Clearly, figure ABC is congruent to
A'B'C’ since they are identical in
size and shape.
7.
EXAMPLE FOUR
• Translate figure ABCD
as follows 9 units to the
left and 1 unit upwards.
• Translate A’B’C’D’ as
follows 1 unit to the
right and 9 units
downward.
8.
In each of the following
diagrams, a point has been
translated by a horizontal
move followed by a vertical
move to form its image.
9.
Describe the translation and then represent the translation
in mathematical notation (algebraically).
• EXAMPLE 1
• Point A moved left by 8 units
and then downwards by 4
units to form A', the image of
A.
The x-coordinate of A' was
obtained by subtracting 8
from the x-coordinate of A.
The y-coordinate of A’ was
obtained by subtracting 4
from the y-coordinate of A.
In other words, the image A'
is the point A'(3-8; 5-4).
We say that A(3; 5) has been
translated by (-8 ; - 4).
Algebraically:
(x;y)⇾(x-8; y-4)
10.
EXAMPLE 2
• Point B moved 6 units right and
then upwards by 4 units to
form B', the image of B.
• The x-coordinate of B’ was
obtained by adding 6 to the x -
coordinate of B.
• The y-coordinate of B’ was
obtained by adding 4 to the y-
coordinate of B.
• In other words, the image B is
the point B‘ (-3 + 6; 5 + 4). We
say that B (-3; 5) has been
translated by (6; 4).
• We say algebraically that B has
been mapped onto B' by the
rule:
(x; y) ⇾(x+6; y+4)
11.
EXAMPLE 3
• Point A did not move
vertically at all. It just moved
5 units to the left.
• The y- coordinate of A' is
the same as A because there
is no vertical movement.
• The x - coordinate of A' was
obtained by subtracting 5
from the x – coordinate of A.
In other words, the image A'
is the point A' (8-5; 4).
• Algebraically:
(x;y)⇾(x-5; y+0)
12.
To summarize:
• We translate the point (x; y) to the point
(x + p; y + q) by a translation of (p ; q)
• Where p is a horizontal move and q is a vertical move.
• If p > 0, the horizontal translation is to the right.
• If p < 0, the horizontal translation is to the left.
• If q > 0, the vertical translation is upward.
• If q < 0, the vertical translation is downward.
13.
1. Determine the coordinates of the image, P’, of the point
P(- 5;-3) if the translation of P to P' is (5; - 6).
14.
2. Represent the translation algebraically if the
point Q (5; 6) is translated to the point Q‘ (- 6; -5).
15.
TRANSLATION OF A FIGURE
• Draw the image A'B'C'D' and
indicate the coordinates of
the vertices of the newly
formed figure.
• The translation here is
(7; - 10), i.e. 7 units to the
right and 10 units
downward.
• The coordinates of ABCD
are as follows: A(-1;3),
B(-6;3), C(-6;7) and D(-1;7)
• First draw ABCD.
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