**TOPIC 7: GEOMETRIC AND TRANSFORMATIONS**

**Reflection**

The Characteristics of Reflection in a Plane

Describe the characteristics of reflection in a plane

transformation in a plane is a mapping which moves an object from one

position to another within the plane. Think of a book being taken from

one comer of a table to another comer. Figures on a plane of paper can

also be shifted to a new position by a transformation. The new postion

after a transformation is called the image. Examples of transformations

are reflection, rotation, enlargement and translation.

Different Reflections by Drawings

Represent different reflections by drawings

A

reflection is a transformation which reflects all points of a plane in a

line called the mirror-line. The image in a mirror is as far behind the

mirror as the object is in front of the mirror

reflection is a transformation which reflects all points of a plane in a

line called the mirror-line. The image in a mirror is as far behind the

mirror as the object is in front of the mirror

Characteristics of Reflection

In

the diagram, APQR is mapped onto ΔP’Q’R’ under a reflection in the line

AB. If the paper is folded along the line AB, ΔPQR will fall in exactly

onto ΔPQR. The line AB is the mirror-line. which is the perpendicular

bisector of PP’, QQ’ and ΔPQR and ΔP’Q’R are congruent.

the diagram, APQR is mapped onto ΔP’Q’R’ under a reflection in the line

AB. If the paper is folded along the line AB, ΔPQR will fall in exactly

onto ΔPQR. The line AB is the mirror-line. which is the perpendicular

bisector of PP’, QQ’ and ΔPQR and ΔP’Q’R are congruent.

Some characteristics observed under reflection are:

- PP’ is perpendicular to AB, RR’ is perpendicular to AB and QQ is perpendicular to AB.
- The image of any point on the Q’ mirror line is the point itself.
- PP’ is parallel to RR’ and QQ’

Reflection in the Line

*y = x*The

line y = x makes an angle 45° with the x and y axes. It is the line of

symmetry for the angle YOX formed by the two axes. By using the

isosceles triangle properties, reflection of the point (1, 0) in the

line y = x will be (0, 1).

line y = x makes an angle 45° with the x and y axes. It is the line of

symmetry for the angle YOX formed by the two axes. By using the

isosceles triangle properties, reflection of the point (1, 0) in the

line y = x will be (0, 1).

The

reflection of (0,2) in the liney = x will be (2,0). You notice that the

co-ordinates are exchanging positions. Generally, the reflection of the

point (a,b) in the line y = x is (b,a).

reflection of (0,2) in the liney = x will be (2,0). You notice that the

co-ordinates are exchanging positions. Generally, the reflection of the

point (a,b) in the line y = x is (b,a).

The reflection of the point B(c,d) in the line y = -x is B’ (-d, -c)

Exercise 1

- Find the image of the point D(4,2) under a reflection in the x-axis.
- Find the image of the point P(-2,5) under a reflection in the x-axis.
- Point Q(-4,3) is reflected in the y-axis. Find the coordinates of its image.
- Point R(6,-5) is reflected in the y-axis. Find the co-ordinates of its image.
- Reflect the point (1 ,2) in the line y = -x.
- Reflect the point (5,3) in the line y = x.
- Find the image of the point (1 ,2) after a reflection in the line y=x followed by another reflection in the line y = -x.
- Find

the image of the point P(-2,1) in the line y = -x followed by another

reflection in the line x = 0 ketch the positions of the image P and the

point P, indicating clearly the lines involved. - Find the co-ordinates of the image of the point A(5,2) under a reflection in the line y = 0.
- Find the coordinates of the image of the point under a reflection in the line x = 0.
- The co-ordinates of the image of a point R reflected in the x axis is R(2, -9). Find the coordinates of R.

Combined Transformations

Draw combined transformations

Combined

Transformation means that two or more transformations will be Performed

on one object. For instance you could perform a reflection and then a

translation on the same point

Transformation means that two or more transformations will be Performed

on one object. For instance you could perform a reflection and then a

translation on the same point

Example 3

What type of transform takes ABCD to A’B’C’D’?

**Solution**

The type of transform takes ABCD to A’B’C’D’ is

**Reflection**Simple Problems on Combined Transformations

Solve simple problems on combined transformations

Exercise 5

What type of transform takes ABCD to A’B’C’D’?

The

transformation ABCD → A’B’C’D’ is a rotation around(-1, 2)by___°.Rotate

P around(-1, 2)by the same angle. (You may need to sketch things out on

paper.)P’ = (__,__)

transformation ABCD → A’B’C’D’ is a rotation around(-1, 2)by___°.Rotate

P around(-1, 2)by the same angle. (You may need to sketch things out on

paper.)P’ = (__,__)

The

transformation ABCD → A’B’C’D’ is a rotation around(-1, -3)by__°Rotate P

around(-1, -3)by the same angle. (You may need to sketch things out on

paper.)P’ = (__,__)

transformation ABCD → A’B’C’D’ is a rotation around(-1, -3)by__°Rotate P

around(-1, -3)by the same angle. (You may need to sketch things out on

paper.)P’ = (__,__)