**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.

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**2. **Under a translation, the shape, size and orientation of object and its image are the same.

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**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 *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
$\left(\begin{array}{l}\text{}4\\ -3\end{array}\right)$

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

.

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
$\left(\begin{array}{l}-4\\ \text{}3\end{array}\right).$

What are the coordinates of point *K*?

Solution**:**

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

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 4**:**

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

State the coordinates of the image of point *A*.

Solution**:**

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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.