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Transformations (I)

11.2.2 Transformations (I), PT3 Focus Practice

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Question 6:
Diagram in the answer space shows a Cartesian plane board used in an indoor game.
The instruction of the game is such that the first move follows the translation ( −4 −2 ) and the second move follows the translation (   5 −4 ) .
Jimmy starts moving from the position mark x.
(a) Mark T1 at Jimmy’s position after the first move and T2 at the position after the second move.
(b) State the coordinates of the position based on Jimmy’s final move.

Answer:


Solution:
(a)




(b) Jimmy’s final position = (3, –2).


Question 7:
In the answer space below, a quadrilateral PQRS is drawn on a grid of squares. A’ is the image of P under rotation 90o at point C.
(a) State the direction of the rotation.
In the answer space,
(b) mark B’ as the image of point Q under the same rotation.
(c) draw the image of quadrilateral PQRS under a reflection on the line MN.

Answer:


Solution:
(a) A’ is the image of P under a rotation of 90o clockwise about the centre C.

(b) and (c)




Question 8:
Diagram in the answer space shows trapezium ABCD drawn on a Cartesian plane. A’D’ is the image of AD under a rotation at centre W.
(a) State
(i) angle of rotation
(ii) the direction of the rotation.
(b) On diagram in the answer space, complete the image of trapezium ABCD.

Answer:



Solution:
(a)(i)
∠DWD'= 90 o  or  270 o .

(a)(ii) Anticlockwise or clockwise.

(b)



11.2.1 Transformations (I), PT3 Focus Practice

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11.2.1 Transformations (I), PT3 Focus Practice

Question 1:
Diagram below in the answer space shows object P drawn on a grid of equal squares with sides of 1 unit.
On the diagram, draw the image of object P under the translation ( − 6 3 ) .   

Answer:

Solution:

Question 2:
Describe the translation which maps point P onto point P’.


Solution:

The translation is ( 7 − 6 ) .  


Question 3:
Diagram below in the answer space shows quadrilateral PQRS. R’S’ is the image of RS under a reflection in the straight line AB.
On diagram in the answer space, complete the image of quadrilateral PQRS.

Answer:

 
Solution:
 

Question 4:
Diagram in the answer space, shows two polygons, M and M’, drawn on a grid of equal squares with sides of 1 unit. M’ is the image of M under a reflection.
(a) Draw the axis of reflection.
(b) Mark the image of P under the same reflection.
(c) Draw the image of M under reflection in the x-axis.

Answer:
 
Solution:



Question 5:
On diagram in the answer space, triangle P’Q’R’ is the image of triangle PQR under a rotation about centre C.
(a) State the angle and direction of the rotation.
(b) K’is the image of point K under the same rotation.
Mark and state the coordinates of K’.

Answer:


Solution:

(a) ∆ P’Q’R’ is the image of ∆ PQR under a clockwise rotation of 90o.
(b) Image of K = (1, –4).


11.1 Transformations (I)

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11.1 Transformations (I)

11.1.1 Transformation
A transformation is a one-to-one correspondence or mapping between points of an object and its image on a plane. 


11.1.2 Translation
1. A translation is a transformation which moves all the points on a plane through the same distance in the same direction.
 
2. Under a translation, the shape, size and orientation of object and its image are the same.
 
3. A translation in a Cartesian plane can be represented in the form ( a b ) ,  whereby, a represents the movement to the right or left which is parallel to the x-axis and b represents the movement upwards or downwards which is parallel to the y-axis.

Example 1:
Write the coordinates of the image of A (–2, 4) under a translation ( 4 − 3 )  and B (1, –2) under a translation ( − 5 3 ) .

Solution:
 
A’ = [–2 + 4, 4 + (–3)] = (2, 1)
B’ = [1 + (–5), –2 + 3] = (–4, 1)

Example 2:
Point K moved to point K’ (3, 8) under a translation ( − 4 3 ) .  
What are the coordinates of point K?

Solution:
K ( x , y ) → ( − 4 3 ) → K ' ( 3 , 8 )
The coordinates of K = [3 – (– 4), 8 – 3]
= (7, 5)

Therefore the coordinates of K are (7, 5).



11.1.3 Reflection
1. A reflection is a transformation which reflects all points of a plane in a line called the axis of reflection.
2. In a reflection, there is no change in shape and size but the orientation is changed. Any points on the axis of reflection do not change their positions.

Example 3:

11.1.4 Rotation
1. A rotation is a transformation which rotates all points on a plane about a fixed point known as the centre of rotation through a given angle in a clockwise or anticlockwise direction.
2. In a rotation, the shape, size and orientation remain unchanged.
3. The centre of rotation is the only point that does not change its position.

Example 4:
Point A (3, –2) is rotated through 90o clockwise to A’ and 180o anticlockwise to A1 respectively about origin.
State the coordinates of the image of point A.

Solution:
Image A’ = (–2, 3)
Image A1 = (–3, 2) 


11.1.5 Isometry
1. An isometry is a transformation that preserves the shape and size of an object.
2.Translation, reflection and rotation and a combination of it are isometries.


11.1.6 Congruence
1. Congruent figures have the same size and shape regardless of their orientation.
2. The object and the image obtained under an isometry are congruent.