**11.1 Transformations (I)**

11.1.1 Transformation

11.1.1 Transformation

**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 $\left(\begin{array}{l}a\\ b\end{array}\right),$ whereby,

*a*represents the movement to the

**right**or

**left**which is

**parallel**to the

**and**

*x*-axis*b*represents the movement

**upwards**or

**downwards**which is

**parallel**to the

**y****-axis**.

Example 1Example 1

**:**

Write the coordinates of the image of

*A*(–2, 4) under a translation $\left(\begin{array}{l}\text{}4\\ -3\end{array}\right)$ and*B*(1, –2) under a translation $\left(\begin{array}{l}-5\\ \text{}3\end{array}\right)$ .

SolutionSolution

**:**

*A*’ = [–2 + 4, 4 + (–3)] = (2, 1)

*B*’ = [1 + (–5), –2 + 3] = (–4, 1)

Example 2Example 2

**:**

Point

*K*moved to point*K*’ (3, 8) under a translation $\left(\begin{array}{l}-4\\ \text{}3\end{array}\right).$What are the coordinates of point

*K*?

SolutionSolution

**:**

$K\left(x,\text{}y\right)\to \left(\begin{array}{l}-4\\ \text{}3\end{array}\right)\to K\text{'}\left(3,\text{}8\right)$

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.

**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 4Example 4

**:**

Point

*A*(3, –2) is rotated through 90^{o}clockwise to*A’*and 180^{o}anticlockwise to*A*respectively about origin._{1}State the coordinates of the image of point

*A*.

SolutionSolution

**:**

Image

*A*’ = (–2, 3)Image

*A*_{1 }= (–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.