Transcript
• TARGET AUDIENCE : GRADES 7-9• DURATION: 1 HOUR
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.
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.
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.
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.
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.
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.
In each of the following diagrams, a point has been translated by a horizontal
move followed by a vertical move to form its image.
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)⇾
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)⇾
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)⇾
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.
1. Determine the coordinates of the image, P’, of the point P(- 5;-3) if the translation of P to P' is (5; - 6).
2. Represent the translation algebraically if the point Q (5; 6) is translated to the point Q‘ (- 6; -5).
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.
Determine the translation rule in each case:
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