Table of Contents

**TOPIC 7: GEOMETRIC AND TRANSFORMATIONS ~ MATHEMATICS FORM 2**

**Reflection**

The Characteristics of Reflection in a Plane

Describe the characteristics of reflection in a plane

A 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**

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.

**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*

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).

**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.

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**

**Reflection**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’ = (__,__)

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’ = (__,__)

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