• 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: