Form 2 Mathematics – GEOMETRICAL TRANSFORMATIONS

GEOMETRICAL TRANSFORMATIONS

-A transformation changes the position, size, direction or shape of objects.
-Transformation in a plane is a mapping which 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

1. Reflection
2. Rotation
3. Enlargement and
4. 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 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: My = x,is given by My = ^–x

A) Reflection in the x-axis

Example

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

Solution:

2

(B)Reflection in y-axis

2. Find the image of 0(3,4) under the reflection in the Y-axis

   Solution:

   Exercise 1

3. 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 area 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 coordinate of the image is A'(-2,3), B'(-2,-4). C'(-4,-4) and D'(-4,3)

1. C) THE REFLECTION IN THE LINE Y = X

The line y = x makes an angle 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

1. 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:
1. M _x-axis P (x,y) = P′ (x, -y)
2. <sub>M y-axis P (x,y) = P′ (-x, y)
3. M Y=x P (x,y) = P′ (y,x)
4.M y=-x P (x,y) = P′ (-y, -x)</sub>

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 -ve and when measured anticlockwise direction the angle is +ve

Example

1. Find the image of the point P(1,0) after a rotation through 90⁰ about the origin in anti-clockwise 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 when it takes (^–2, ^–3)
   Solution

T (2,^–3) = (0, ^–8)

2. Find the image of the point (1,2) under a rotation through 180⁰ anti-clockwise about the origin
   Solution
3. Find the rotation of the point (6, 0) under a rotation through 90⁰ clockwise about the origin
   Solution
4. Find the image of (1,2) after a rotation of ^–90⁰ ant –clock wise
   Solution
5. Find the image of (^–3, 5) after a rotation of ^–180^{0
   Solution}
   

6. The vertices of rectangle PQRS are P(0,0), Q (3,0), R (3,2), S (0, 2). The rectangle is rotated through 90⁰ clockwise about the origin.

(a) Find the co-ordinates of its image

(b) Draw the image

More examples on translation

1. Translation takes the origin to (-2, 5)

Find where it takes

(a) (-6, 6)

(b) (5, 4 )

Solution

(a)= =

= =

=

= +

=

: The translation takes (-6,6) to

(b) = +

=

: 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

(a) (0,0)

(b) (1,1)

(c) (3,7)

Solution

(a).

= +

= +

=

: . The translation takes the origin to (^–1, ^–2)

(b). = +

= +

=

3. 3. A translation moves the origin a distance 2 units along the line y= x upwards.

Find where it takes

(a) (0,0)

(b) (2, ^–1)

(c) (1, 1)

Solution

x = 2cos 45⁰ = 2 x =

Sin 45⁰ = =

y = 2sin 45⁰ = 2 x =

Translation factor (, )

(a). = +

=

: . The origin is translated to ( , )

(b). = + =

: . (2, ^–1) is translated to ( (+ 2), (- 1))

4. A translation takes the point

(3, 2) to (-4, -5), Find where it takes

(0 , 0)

Solution

= + where is translation factor

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 and if it is greater than 1 the figure is enlarged K times.

– In 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

= x 3 x 3
= 4.5 square units

The area of the new triangle = 4.5 x K²

= 4.5 x 2²

= 18 square units

3. The line segment AB with coordinated A (4,0) and B (0,3) enlarge 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 (5,5)

EXERCISE 2

(a) Δ ADE to Δ ABC?

(b) Δ ADE to Δ AFG?

(a) 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)

2. ABCD is a parallelogram

Solution

EXERCISE 3

1. List 3 examples of isometric transformation

o Translation
o Rotation
o 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 90⁰ about the

R_90º(6, ^–8) = (^–6,8)

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 wich moves the origin to (2,4)

Solution

A = = +

A =

A = (4,9)

B = = +

=

B = (7, 9)

C = = +

=

C = (8, 12)

D = + =

D = (5, 12)

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