ICSE Class 10 Maths Chapter 12 Reflection

Unit 3: Co-ordinate Geometry

Chapter 12: Reflection

In x-axis, y-axis, x = a, y = a and the origin; Invariant Points

12.1 Introduction

Co-ordinate geometry is the branch of geometry in which two numbers, called co-ordinates, are used to locate the position of a point in a plane.

12.2 Co-ordinate Axes

The two mutually perpendicular number lines intersecting each other at their zeroes, are called rectangular axes or co-ordinate axes or axes of reference.

As shown in the adjacent figure, the horizontal number line XOX' is called the x-axis; the vertical number line YOY' is called the y-axis and their point of intersection, O is called the origin.

12.3 Co-ordinates

The position of a point in a plane is expressed by a pair of two numbers (one concerning x-axis and the other concerning y-axis) called co-ordinates.

Consider a point P (x, y)

Here (x, y) is a pair of two numbers, which gives the co-ordinates of point P.

The first number x of the pair (x, y) is the distance of the point P from y-axis and is called x-co-ordinate or abscissa.

The second number y of the pair (x, y) is the distance of the point P from x-axis and is called the y-co-ordinate or ordinate.

Suppose the co-ordinates of point A are (-3, 4), then its abscissa = -3 and ordinate = 4.

And, if for a point B, abscissa = -2 and ordinate = -3, then its co-ordinates are (-2, -3).

Remember:

In stating the co-ordinates of a point the abscissa precedes the ordinate. The two co-ordinates are separated by a comma and are enclosed in a bracket. Thus, a point with abscissa x and ordinate y is denoted by (x, y).

Co-ordinates of origin O = (0, 0).

Co-ordinates of a point on the x-axis = (x, 0) and

Co-ordinates of a point on the y-axis = (0, y).

12.4 Reflection

When an object is placed before a plane mirror, the image formed is at the same distance behind the mirror as the object is in front of it.

Therefore, to find the image of a point P in a line AB, consider AB as the plane mirror and point P as the object. Now, find a point P' on the other side of AB, such that P' is at the same distance from AB as P is from it.

Thus, point P' is the image of point P in line AB and line AB, which is also the perpendicular bisector of PP', is said to be the mirror line or mediator of segment PP'.

The transformation which maps a point P to P' is called reflection.

The reflection can be denoted in several ways, but here it will be denoted by M_l, where M denotes reflection and l is the line or point in which the reflection takes place.

Thus, M_x represents reflection in the x-axis;

M_y represents reflection in the y-axis;

and M_o represents reflection in the origin.

12.5 Reflection in the line y = 0 i.e. in the x-axis

The line y = 0 means the x-axis

The adjoining figure shows the reflection of point P(x, y) in the x-axis. It is clear from the figure that P' is the image of P in the x-axis such that P' = (x, -y).

Symbolically, M_x (x, y) = (x, -y).

Therefore, when a point is reflected in the x-axis, the sign of its ordinate changes.

For Example:

Reflection of point (2, 3) in the x-axis = (2, -3) i.e. M_x (2, 3) = (2, -3)

Reflection of point (2, -3) in the x-axis = (2, 3) i.e. M_x (2, -3) = (2, 3)

Similarly, M_x (-5, 7) = (-5, -7); M_x (-a, -b) = (-a, b) and so on.

12.6 Reflection in the line x = 0 i.e. in the y-axis

The line x = 0 means the y-axis

As is clear from the adjoining figure, the reflection of point P (x, y) in the y-axis is P' such that P' = (-x, y).

Symbolically, M_y (x, y) = (-x, y)

Therefore, when a point is reflected in the y-axis, the sign of its abscissa changes.

For Example:

Reflection of point (2, 3) in the y-axis = (-2, 3) i.e. M_y (2, 3) = (-2, 3)

Reflection of point (2, -3) in the y-axis = (-2, -3) i.e. M_y (2, -3) = (-2, -3)

Similarly, M_y (-5, 7) = (5, 7), M_y (-a, -b) = (a, -b) and so on.

12.7 Reflection in the origin

When a point P (x, y) is reflected in the origin, the signs of its abscissa and ordinate both change i.e. if P' is the image of P (x, y) in the origin, then P' = (-x, -y)

Symbolically, M_o (x, y) = (-x, -y)

For Example:

Reflection of point (2, 3) in the origin = (-2, -3) i.e. M_o (2, 3) = (-2, -3)

Reflection of point (2, -3) in the origin = (-2, 3) i.e. M_o (2, -3) = (-2, 3)

Similarly, M_o (-5, 7) = (5, -7); M_o (-a, -b) = (a, b) and so on.

Example Problem 1

The triangle A(1, 2), B(4, 4) and C(3, 7) is first reflected in the line y = 0 onto triangle A'B'C' and then triangle A'B'C' is reflected in the origin onto triangle A''B''C''. Write down the co-ordinates of:

(i) A', B' and C'

(ii) A'', B'' and C''.

Solution:

Reflection in y = 0 means reflection in x-axis.

(i) Since, reflection in the x-axis is given by M_x (x, y) = (x, -y)

A' = reflection of A (1, 2) in the x-axis = (1, -2)

Similarly, B' = (4, -4) and C' = (3, -7)

(ii) Since, reflection in the origin is given by M_o (x, y) = (-x, -y)

A'' = reflection of A' (1, -2) in the origin = (-1, 2)

Similarly, B'' = (-4, 4) and C'' = (-3, 7)

Example Problem 2

A point P is reflected in the x-axis. Co-ordinates of its image are (8, -6).

(i) Find the co-ordinates of P.

(ii) Find the co-ordinates of the image of P under reflection in the y-axis.

Solution:

(i) P = (8, 6) Since, M_x (8, 6) = (8, -6)

(ii) Co-ordinates of the image of P(8, 6) under reflection in the y-axis = (-8, 6)

Example Problem 3

Perform the operations M_x . M_y and M_y . M_x on the point (3, -4). State whether M_x . M_y = M_y . M_x. If 'yes'; then state whether it is always true.

Solution:

M_x . M_y (3, -4) = M_x [M_y (3, -4)] = M_x (-3, -4) = (-3, 4)

M_y . M_x (3, -4) = M_y [M_x (3, -4)] = M_y (3, 4) = (-3, 4)

Therefore M_x . M_y = M_y . M_x

'Yes', it is always true.

Remember:

The combination of the reflections is always commutative, i.e.

(i) M_x . M_y = M_y . M_x = M_o

(ii) M_o . M_x = M_x . M_o = M_y

(iii) M_o . M_y = M_y . M_o = M_x and so on.

12.8 Invariant Point

Any point that remains unaltered under a given transformation is called an invariant.

e.g. when the point A(5, 0) is reflected in the x-axis, the co-ordinates of its image are also (5, 0) i.e. the co-ordinates remain unchanged.

Therefore, Point A (5, 0) is said to be invariant under reflection in the x-axis.

The same is with:

(i) B (0, 5) in invariant under reflection in the y-axis.

(ii) O (0, 0) in invariant under reflection in the x-axis, y-axis and origin.

(iii) C (-3, 0) in invariant under reflection in the x-axis and so on.

Remember:

In case of an invariant point, the point is its own image i.e. reflection of the point is the point itself. Such transformation (reflection) is called invariant transformation. Similarly, reflection of any point is invariant under reflection in a line, if the point lies in the same line.

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