Transforming a figure is changing its size, location, orientation or shape.
The sides of the figure may change but they will remain in proportion to the original.
The sizes of the angles do not change.
Transforming a figure can involve:

• translations
• reflections
• rotations
• enlargements
• reductions

A glide reflection is a combined transformation consisting of a reflection followed by a translation parallel to the line of reflection or vice versa.

Assessment Standards

Investigate , generalize and apply the effect of the following transformations of the point (x;y):

• a translation of p units horizontally and q units vertically
• a reflection in the x –axis , y-axis or the line y = x.

Example 1
1.1. P(-2;-3) to P'(3;1) has been translated according to the rule:

(x;y) → (x+5;y+4) where P has moved 5 units horizontally to the right,
and 4 units vertically upwards.

Translating a figure does not change the size of the figure.
The image is congruent to the original figure.

Example 2

2.1. The image of P(2;3), after the following translations.
2.1.1 (x;y) → (x-3;y+1) will become
P'(2-3; 3+1) = P'(-1;4)

2.1.2 (x;y) → (x+6; y-4) will become
P'(2+6; 3-4) = P'(8; -1)

Tutorial

1. Determine the following translations. Give answers in the form:
(x;y) → (x+a;y+b)

1.1. P(0;3) to P’(7;5)
1.2. Q(3;-3) to Q’(-1;-1)

2. Determine the image of P’of P(3;4) after each of the following translations.

2.1. (x;y) → (x-8;y+3)
2.2. (x;y) → (x+2;y-7)

3. The point A is reflected about a line to give the image of A’.
In each case , give the line of reflection.

3.1. A(3;4) to A’ (4;3)
3.2. A(3;4) to A’(3;-4)
3.3. A(4;5) to A’ (-4;5)
3.4. A(2;3) to A’ (3;2)

4. Write down the co-ordinates of the image when the following points are reflected according to the given rule:

4.1. P(-4;3) : (x;y) → (-x;y)
4.2. Q(-5;1) : (x;y) → (x;-y)
4.3. R(-4;3) : (x;y) → (y;x)
4.4. Q(-2;-6) : (x;y) → (-x;-y)

5.1. Draw on the graph paper provided in the diagram sheet, the pentagon whose vertices have the following co-ordinates:

    A(-2;3),
    B(3;1),
    C(1;-4),
    D(-3;-5)
    E(-5;-1)

Now construct, on the same system of axes, the pentagon formed by shifting ABCDE 2 units upwards and 1 unit to the right.
Indicate the co-ordinates of the vertices.

5.2. P(-4;1), Q(2;3), R(8;9) and S(2;7) are four points in a Cartesian plane.
Answer the questions that follow:

5.2.1. Show that the co-ordinates of T, the midpoint of the diagonals PR and QS is (2; 5)

5.2.2. If PQRS is moved such that the co-ordinates of T become T(5;5), describe the shift of the parallelogram.

5.2.3. Write down the new co-ordinates of P’ and S’.