MATHEMATICS O LEVEL(FORM TWO) NOTES – GEOMETRICAL TRANSFORMATIONS

GEOMETRICAL TRANSFORMATIONS

A transformation changes the position, size, direction, or shape of objects.

Transformation in a plane is a mapping that moves an object from one position to another within the plane. The new position after a transformation is called an image.

Examples of transformations are

• Reflection
• Rotation
• Enlargement
• Translation

Suppose a point P[x, y] in the xy-plane moves to a point P’ [x’, y’] by a transformation T.

P is said to be mapped to P’ by T and may be indicated as:

A transformation in which the size of the image is equal to the size of the object is called an Isometric mapping.

REFLECTION

Reflection is an example of an isometric mapping.

Isometric mapping means the distance from the mirror to an object is the same as that from the mirror to the image.

The plane mirror is the line of symmetry between the object and the image.

The line joining the object and the image is perpendicular to the mirror.

NOTE

• The symbol/letter for reflection is M.
• The reflection in the X-axis and Y-axis are indicated as M_x and M_y respectively.
• The reflections in lines with certain equations are indicated with their equations as subscripts.

For example: M_y = x is given by M_y = -x.

A) Reflection in the x-axis

Example

1. Find the image of the point A(2,1) after a reflection in the x-axis.

Solution:

B) Reflection in y-axis

2. Find the image of O(3,4) under the reflection in the Y-axis.

Solution:

Exercise 1

1. Find the image of the point D(4,2) under a reflection in the x-axis.

Solution:

2. Find the image of the point P(-2,5) under the reflection in the x-axis.

Solution:

3. Point Q (-4,3) is reflected in the Y-axis.

Solution:

4. Point R (6, 5) is reflected in the X-axis. Find the coordinates of its image.

5. The vertices of a triangle PQR are P (6, 2), Q (-2, 8), R (-5, -1). If triangle PQR is reflected in the Y-axis, find coordinates of the vertices of its image.
6. The vertices of rectangle ABCD are A (2,3), B (2,-4), C (4, -4), D (4,3). Rectangle ABCD is reflected in the Y-axis.

(a) Find the coordinates of the vertices of its image.

(b) Draw a sketch to show the image.

Solution:

6(a) The coordinates of the image are A'(-2,3), B'(-2,-4), C'(-4,-4), and D'(-4,3).

C) Reflection in the line y = x

The line y = x makes an angle of 45º with the x and y axes.

See the diagram:

∴ M_y=x (x,y) = (y,x)

Example

1. Find the image of point A(1,2) after a reflection in the line y=x.

D) Reflection in the line y = -x

∴ M_y=-x (x,y) = (-y,-x)

Example

Find the image of B (-3, 4) after a reflection in the line y=-x followed by another reflection in the line y=0.

Solution

The reflection of B (-3, 4) in the line y = -x is B’ (-4, 3) and the image of B’ (-4, 3) after reflection in the line y = 0 is B’ (-4, -3).

NOTE:

If P is the object, the reflection of point P(x,y) will be:

• M_x-axis: P(x,y) = P′(x, -y)
• M_y-axis: P(x,y) = P′(-x, y)
• M_y=x: P(x,y) = P′(y, x)
• M_y=-x: P(x,y) = P′(-y, -x)

Rotation

• Rotation is a transformation which moves a point through a given angle.
• The angle turned through can be either in clockwise or anticlockwise direction.
• Rotation is an isometric mapping and usually denoted as R. R_θ means a rotation through an angle θ.
• In the XY plane, when θ is measured in the clockwise direction, the angle is negative; when measured anticlockwise, the angle is positive.

Example

1. Find the image of the point P(1,0) after a rotation through 90° about the origin in anticlockwise direction.

TRANSLATION

• Translation is a straight movement without turning.
• A translation is usually denoted by T. For example, T(1,1) = (6,1) means that the point (1,1) has been moved to (6,1) by a translation T.
• This translation will move the origin (0,0) to (5,0) and it is written as T = (5/0).

Examples:

1. A translation takes the origin to (-2, -5). Find where it takes (-2, -3) and (5, 4).

Solution:

(a) T(-6, 6) = (0, -8)

The translation takes (-6, 6) to (0, -8).

(b) T(5, 4) = (3, 9)

The translation takes (5, 4) to (3, 9).

2. A translation takes every point a distance of 1 unit to the left and 2 units downwards on the xy-plane. Find where it takes (0,0), (1,1), and (3,7).

Solution:

(a) The origin is translated to (-1, -2).

(b) The point (1,1) is translated to (0, -1).

(c) The point (3,7) is translated to (2, 5).

3. A translation moves the origin a distance 2 units along the line y = x upwards. Find where it takes (0,0), (2,-1), and (1,1).

Solution:

ENLARGEMENT

Enlargement is a transformation in which a figure is made larger (magnified) or made smaller (diminished).

• The number that magnifies or diminishes a figure is called the enlargement factor, usually denoted by letter K. If K is less than 1, the figure is diminished; if it is greater than 1, the figure is enlarged K times.
• In the case of closed figures, if the lengths are enlarged by a factor K, then the area is enlarged by K².

Examples:

1. Draw a triangle PQR with vertices P (0,0), Q (0, 3), and R (3, 0).

P’ = 2 (0,0) = (0,0)

Q’ = 2 (0,3) = (0,6)

R’ = 2 (3,0) = (6,0)

2. From the above question, what is the area of the new (enlarged) triangle?

Solution:

Area of the original triangle = ½ × 3 × 3 = 4.5 square units.

The area of the new triangle = 4.5 × K² = 4.5 × 2² = 18 square units.

3. The line segment AB with coordinates A (4,0) and B (0,3) enlarges to A’B’ by a factor 2. Find the coordinates for A’ and B’.

A’ = 2 (4, 0) = (8,0)

B’ = 2 (0, 3) = (0,6)

4. Find the image of the circle of radius one unit having its centre at (1,1) under enlargement transformation factor 5.

Solution:

= 5(1,1) = (5,5)

The image of the enlarged circle is at (5,5).

EXERCISE 2

1. Δ ADE to Δ ABC?
2. Δ ADE to Δ AFG?

Solution:

Δ ADE to Δ ABC =

2. The point P(6,2) is enlarged by factor of 4, what is the new end point?

Solution:

4 (6,2) = (24, 8)

The point is (24, 8)

3. The vertices of parallelogram ABCD are given. Find the image after enlargement.

Solution:

EXERCISE 3

1. List 3 examples of isometric transformation.

• Translation
• Rotation
• Reflection

2. Is enlargement an Isometric transformation?

Enlargement is not an Isometric transformation.

3. Find the image of the point Q (6, -8) after a rotation of 90° about the origin.

Draw a parallelogram ABCD with vertices A (2,5), B (5,5), C (6,8), D (3,8). Find and draw the image parallelogram formed by the translation which moves the origin to (2,4).

Solution:

A = (4,9)

B = (7,9)

C = (8,12)

D = (5,12)