ICSE Class 10 Maths Chapter 12 Reflection

Unit 3 Co-ordinate Geometry Chapter 12 Reflection

Points To Remember

Co-ordinate Axes

The position of a point in a plane is determined by two fixed mutually perpendicular straight lines X' OX and YOY', intersecting each other at a point O.

These lines are called the co-ordinate axes or Axes of Reference.

The horizontal line X'OX is called the x-axis.

The vertical line YOY' is called the y-axis.

The point O is called the origin.

We fix up a convenient unit of length and starting from the origin as zero, mark off equal distances on x-axis as well as y-axis.

The distances measured along OX and OY are taken as positive, while those along OX' and OY' are taken as negative, as shown in the adjoining figure.

Co-ordinates of a Point

Let P be a point in a plane.

Let, the distance of P from y-axis = a units.

And, the distance of P from x-axis = b units.

Then, we say that the co-ordinates of P are (a, b).

a is called the x-co-ordinate or abscissa of P.

b is called the y-co-ordinate or ordinate of P.

We say that P (a, b) is a point.

Distance of any point on x-axis from x-axis is 0.

Co-ordinates of each point on x-axis are (x, 0).

Distance of any point on y-axis from y-axis is 0.

Co-ordinates of each point on y-axis are (0, y).

Teacher's Note

Understanding coordinate axes is fundamental to many real-world applications, from GPS navigation to video game graphics. Every location on Earth can be pinpointed using coordinates, making this concept essential for modern technology.

Equation of Lines

About the equation of a line, we shall study in the next chapter. However, remember the following.

(i) The line x = 0 means y-axis.

(ii) The line y = 0 means x-axis.

(iii) The line x = a means the line parallel to y-axis at a distance a from it.

(iv) The line y = b means the line parallel to x-axis at a distance b from it.

Reflection

Image of An Object In a Mirror

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

Image of a Point in a Line

For finding the image of a point P in a line AB, we consider the line as a plane mirror and P as the object. Now, we 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, the image of a point P in a line AB is a point P' such that AB is the perpendicular bisector of PP'.

Thus, AB perpendicular PP' and if PP' cuts AB in M, then PM = MP'.

Image of a Point in a Point

The image of a point P in a point M is a point P' such that M is the mid-point of PP'.

Reflection: The transformation R subscript l which maps a point P to its image P' in a given line (or point) l, is called a reflection in l.

Thus, R subscript l (P) = P'.

We shall represent:

(i) Reflection in x-axis by R subscript x;

(ii) Reflection in y-axis by R subscript y;

(iii) Reflection in the origin by R subscript o.

(a) Reflection in x-axis

Let P(x, y) be a point in a plane. Draw PM perpendicular OX, meeting it at M.

Produce PM to P' such that MP = MP'.

Then, P' is the image of P when reflected in x-axis.

Clearly, the co-ordinates of P' are P' (x, -y).

P (x, y) when reflected in x-axis, has the image P' (x, -y).

R subscript x (x, y) = (x, -y).

Teacher's Note

Reflections in mirrors and water are everyday examples of geometric reflection. Understanding how reflections work mathematically helps explain natural phenomena and is crucial in designing optical instruments.

(b) Reflection in y-axis

Let P (x, y) be a point in a plane.

Draw PN perpendicular OY meeting it at N.

Produce PN to P' such that NP' = NP.

Then, P' is the image of P when reflected in y-axis.

Clearly, the co-ordinates of P' are P' (-x, y).

P (x, y) when reflected in y-axis, has the image P' (-x, y).

R subscript y (x, y) = (-x, y)

(c) Reflection in the Origin

Let P (x, y) be a point in a plane.

Join PO and produce it to P' such that OP' = OP.

Then, P' is the image of P when reflected in the origin.

Clearly, the co-ordinates of P' and P' (-x, -y).

P (x, y) when reflected in the origin, has the image P' (-x, y).

R subscript o (x, y) = (-x, y)

Invariant Points

A point P is said to be invariant with respect to a given line l, if the image of P in the line l is P itself. This happens when P lies on the line l.

Combination of Reflections

(i) Reflection of P (x, y) in y-axis followed by reflection in x-axis. We denote the combined transformation by R subscript x R subscript y and operate it as under.

(R subscript x R subscript y) (x, y) = R subscript x [R subscript y (x, y)]

= R subscript x (-x, y)

= (-x, -y) = R subscript o (x, y). [Since R subscript x (-x, y) = (-x, -y)]

Therefore, R subscript x R subscript y = R subscript o

(ii) Reflection of P (x, y) in x-axis followed by reflection in y-axis: We denote it by R subscript y R subscript x and in a manner similar as above, we can show that:

R subscript y R subscript x = R subscript o

Thus = R subscript x R subscript y = R subscript y R subscript x = R subscript o

(iii) Reflection of P (x, y) in x-axis followed by reflection in origin: Clearly, it will be denoted by R subscript o R subscript x and we have

(R subscript o R subscript x) (x, y) = R subscript o [R subscript x (x,y)]

= R subscript o (x, -y)

= (-x, y) = R subscript y (x, y).

Therefore, R subscript o R subscript x = R subscript y

(iv) Reflection of P (x, y) in y-axis followed by reflection in origin: The combined reflection is R subscript o R subscript y and in a manner similar as above, we can show that:

R subscript o R subscript y = R subscript x

Similarly, R subscript x R subscript o = R subscript y

and R subscript y R subscript o = R subscript x

Teacher's Note

Combination of reflections appears in kaleidoscope patterns and symmetrical art designs. Understanding how multiple reflections combine helps explain the beautiful patterns found in nature and assists in creating digital graphics and animations.

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