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# Transformation Geometry

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### Transcript of "Transformation Geometry"

1. 1. • TARGET AUDIENCE : GRADES 7-9 • DURATION: 1 HOUR
2. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 14. 2. Represent the translation algebraically if the point Q (5; 6) is translated to the point Q‘ (- 6; -5).
15. 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.
16. 16. Determine the translation rule in each case:
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