11.1 Transformations (I)
A transformation is a one-to-one correspondence or mapping between points of an object and its image on a plane.
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
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.
Write the coordinates of the image of A (–2, 4) under a translation
and B (1, –2) under a translation
A’ = [–2 + 4, 4 + (–3)] = (2, 1)
B’ = [1 + (–5), –2 + 3] = (–4, 1)
Point K moved to point K’ (3, 8) under a translation
What are the coordinates of point K?
The coordinates of K = [3 – (– 4), 8 – 3]
= (7, 5)
Therefore the coordinates of K are (7, 5).
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.
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.
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.
Image A’ = (–2, 3)
Image A1 = (–3, 2)
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.
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.