Reflection

A reflection is a transformation that flips a figure across a line. The line work as a plane mirror. In reflection, the line joining the object and the image is perpendicular to the mirror line. It means the mirror line is perpendicular bisector of the line segment joining object and image. The mirror line is also called the axis of reflection.

Characteristics of reflection of geometrical figures in the axis.

When geometrical figures are reflected in the axis of reflection, the following properties are found.

a. Coordinates can be used for finding images of geometrical figures after the reflection in the lines like X- axis, Y- axis, a line parallel to X- axis, a line parallel to Y- axis, the line y = x, the line y = -x, etc.The distance of the object from the axis of reflection is equal to the distance of reflection is equal to the distance of the image from the axis is a reflection.

OP = OP' as shown in fig 1.

b. The shape of objects and images are laterally inverted. It means top remains at the top, bottom remains at the bottom but left side goes to the right side and right side goes to the left side as shown in fig 2.

c. The lines joining the same ends of the object and image are perpendicular to reflecting axis.
Axis of reflection is the perpendicular bisector of the line segment joining same ends of object and image.
XX'is perpendicular bisector of AA', BB' and CC' as in fig 3.

d. The points on the axis of reflection are invariant points.

e. Use of co-ordinates in Reflection.

f. In reflection, the object figure and its image figure are congruent to each other.

1. Reflection in the X- axis
   Equation of X- axis is y = 0. So, reflection in X- axis means reflection in the line y = 0. Let P(x, y) be any point in the plane. Draw a perpendicular PL from the point P to the X- axis and produce it to the point P' such that PL = LP'. Then P' is the image of P after reflection in X- axis.
   

   

   
   

   
   Here, co-ordinates of L are (x, 0). Let the co- ordinates of P' be (x', y'). Since L is the mid- point of line segment PP', then by mid- point formula,
   

   x = \(\frac {x+x'}2\)	and	0 = \(\frac {y+y'}2\)
   or, x + x' = 2x	and	y + y' = 0
   or, x' = x	and	y' = -y

   ∴ Co-ordinates of the P' are (x. -y)
   ∴ Image of point P(x, y) after reflection in X- axis is P'(x, -y).
   Hence, if R_x denotes the reflection in X- axis, then:
   Rx: P (x, y)→ P' (x, -y)
   
   
2. Reflection in the Y- axis
   Equation of Y- axis is x = 0. So, reflection in Y- axis means reflection in the line x = 0.
   Let P (x, y) be any point in the plane. Draw a perpendicular PM from the point P to the Y- axis and produce it to the point P' such that PM = MP'. Then P' is the image of P after reflection in Y- axis.
   
   Here, co-ordinate of M is (0, y). Let the co- ordinates of P' be (x', y'). Since M is the mid-point of line segment PP', then by mid- point formula,
   

   0 = \(\frac{x + x'}2\)	and	y = \(\frac {y + y'}2\)
   or, x + x' = 0	and	y + y' = 2y
   or, x' = -x	and	y' = y

   ∴ Co- ordinates of the point P' are (-x, y).
   ∴ Image of point P(x, y) after reflection in Y- axis P' (-x, y).
   Hence, if R_y denotes the reflection in Y- axis, then:
   Ry: P (x, y)→ P' (-x, y)
   
   
   
3. Reflection in the line parallel to X- axis
   The equation of a line parallel to X- axis is given by y = k where k is Y- intercept of the line. So, reflection in the line parallel to X- axis means reflection in the line y = k.
   Let P (x, y) be any point in the plane. Draw a perpendicular PM from P to the line y = k and produce it to the point P' such that PM = PM'. Then P' is the image of P after reflection in

   

   the line y = k.
   Here, co- ordinates of M are (x, k). Let thye co- ordinates of P' be (x', y'). Since, M is the mid-point of the line segment PP', then by mid- point formula,

   x = \(\frac{x + x'}2\)	and	k = \(\frac{y + y'}2\)
   or, x + x' = 2x	and	y + y' = 2k
   or, x' = x	and	y' = 2k - y

   ∴ Co- ordinates of the point P' are (x, 2k - y).
   ∴ Image of point P (x, y) after reflection in the line y = k is P' (x, 2k - y).
   Hence, if R denotes the reflection in the line y = k, then:
   R: P (x, y)→ P' (x, 2k - y)
   
   
4. Reflection in the line parallel to Y- axis
   The equation of a line parallel to Y- axis is given by x = k where k is X - intercept of the line. So, the reflection in a line parallel to Y- axis means reflection in the line x = k.
   Let P (x, y) be any point in the plane. Draw a perpendicular PM from P to the line x = k and produce it to the point P' such that PM = MP'. Then P' is the image of P after reflection in the line x = k.
   

   

   Here, co- ordinates of M are (k, y).
   Let the co- ordinates of P' be (x', y').
   Since, M is the mid- point of the line segment PP', then by mid- point formula,
   

   k = \(\frac {x + x'}2\)	and	y = \(\frac {y + y'}2\)
   or, x + x' = 2k	and	y + y' = 2y
   or, x' = 2k - x	and	y' = y

   ∴ Co- ordinates of the point P' are (2k - x, y).
   ∴ Image of point P (x, y) after reflection in the line x = k is P' (2k - x, y).
   Hence, if R denotes the reflection in the line x = k, then:
   R: P (x, y)→ P' (2k - x, y)
   
   
5. Reflection in the line y = x
   y = x is the equation of the line which makes an angle of 45° with the positive direction of X- axis. Then, slope of the line, m₁ = tan 45° = 1.
   Let P (x, y) be any point in the plane. Draw perpendicular PM from the point P to the line y = x and produce it to the point P' such that PM = MP'. Then P' is the image of the point P under reflection about the line y = x.
   Let (x', y') be the co- ordinates of the point P'.
   Then, slope of the line segment PP', m₂ = \(\frac {y - y'}{x - x'}\).
   Since, the line segment PP' and the line y = x are perpendicular to each other. Then:
   

   

   
   m₁× m₂ = -1
   or, 1× (\(\frac {y-y'}{x-x'}\)) = -1
   or, y - y' = -x + x'
   or, y + x = y' + x' .......................(i)
   Again,
   Co- ordinates of mid- point of the line segment PP' are: (\(\frac {x + x'}2\), \(\frac {y + y'}2\)).
   This point lies in the line y = x.
   So,
   \(\frac {y + y'}2\) = \(\frac {x + x'}2\)
   or, y + y' = x + x'
   or, y - x = x' - y' .........................(ii)
   Adding (i) and (ii) we get, 2y = 2x' or x' = y
   Subtracting (ii) from (i) we get, 2x = 2y' or y' = x.
   ∴ Co- ordinates of the point P' are (y, x).
   ∴ Image of the point P (x, y) under the reflection about the line y = x is the point P (y, x).
   Hence, if R denotes the reflection in the line y = x, then:
   R: P (x, y)→ P' (y, x)
   
   
6. Reflection in the line y = -x
   y = -x is an equation of the line which makes an angle of 135° with the positive direction of X- axis.
   ∴ Slope of the line y = -x is m₁ = tan 135° = -1
   Let P (x, y) be any point in the plane. Draw perpendicular PM from the point P to the line y = -x and produce it to the point P' such that PM = MP'.
   Let the co- ordinates of the point P' be (x', y').
   Slope of the line segment PP' is m₂ = \(\frac {y - y'}{x - x'}\).
   Since, the line segment PP' and the line y = -x are perpendicular to each other, then:
   

   

   m₁× m₂ = -1
   or, (-1)× \(\frac {y - y'}{x - x'}\) = -1
   or, y - y' = x - x' .........................(i)
   Again,
   Co- ordinates of the mid- point of line segment PP' are: (\(\frac {x + x'}2\), \(\frac {y + y'}2\)).
   This point lies in the line y = -x.
   So,
   (\(\frac {y + y'}{2}\)) = - (\(\frac {x + x'}2\))
   or, y + y' = -x' - x .......................(ii)
   Adding (i) and (ii) we get, 2y = - 2x' or x' = -y
   Subtracting (ii) from (i) we get, -2y' = 2x or y' = -x
   ∴ Co- ordinates of P' are (-y, -x)
   ∴ Image of the point P (x, y) after reflection in the line y = -x is P' (-y, -x).
   Hence, if R denotes in the line y = -x, then:
   R : P (x, y)→ P' (-y, -x)