Reflection
Reflection: The process of obtaining its image in the plane mirror is called reflection.
If m is the plane mirror, then
(i) P' is the image of P under reflection in M and NP = NP'.
(ii) NP' is the image of NP under reflection.
(iii) PP' is perpendicular to M and M is perpendicular bisector or mediator of PP'.
(Mediator is another name for perpendicular bisector).
(A) Point reflection: The reflection of A through the point 0 is the point A' such that O is the mid-point of AA'. The point O is called centre of reflection.
INVARIANT POINTS, LINES AND LINE SEGMENTS
(B) Reflection in x-axis: To get the image of a point after its reflection in the x-axis, we change the sign of the x coordinate but keep the y co-ordinate the same
i.e., My (x, y) = (- x, y)
(i) Under reflection in the x-axis, all points on the x-axis are invariant i.e., they are their own image.
Under reflection in the x-axis, lines invariant are:
(a) the x-axis,
(b) the line perpendicular to the x-axis: Line segments right bisected by the x-axis are invariant under reflection in' the x-axis,
(C) Reflection in y-axis: To find the image of any point after reflection in the y-axis, we should retain the x co-ordinate and change the sign of the y co-ordinate.
i.e., Mx (x, y) = (x, - y)
(ii) Under reflection in the y-axis, all points on the y-axis are invariant.
Under reflection in the y-axis, lines invariant are:
(a) the y-axis.
(b) All lines perpendicular to the y-axis (parallel to x-axis): Line segments right bisected by the y-axis are invariant under reflection in the origin.
(D) Reflection in the origin: To get the image of a point after its reflection in the origin, we change the signs of both x co-ordinate and y co-ordinate.
(iii) The only point, invariant under reflection in the origin is the origin itself. rm denote the operation 'reflection in m' and is written 'as rm: P → P' or rm (P) = P'.
The point P is called the pre-image of the point P'.
Note:
(1) x-co-ordinate or abscissa of all the points on the y-axis is zero. Thus, any point on the y-axis is of the form (0. y).
(2) y-co-ordiante or ordinate of all the points on the x-axis, is zero. Thus, any points on the x-axis is of the form (x, 0).
(3) Co-ordinates of origin are (0, 0) as t ms on both the axes.
RFLECTION CONSTRUCTION:
Reflection of a point in a line:
Steps of Construction:
1. Draw PQ perpendicular to AB, meeting AB at O.
2. From OQ cut of OP' = OP.
Then P' is the reflection of P in AB.
REFLECTION OF A LINE SEGMENT IN A LINE
Steps of Construction:
1. Draw PO perpendicular to AB.
Produce PO and cut of OP' = PO.
\ P' is the reflection of P in AB.
2. Similarly, get a', the image of a in AB.
3. Join P' and Q'.
Then P'O' is the reflection of line segment PQ in AB.
Reflection of a point in a point: Let o be the point in which we require to find the reflection of point P.
Steps of Construction:
Join PO and produce it to P' such that PO = OP'
Then P' is the reflection of P in the point O.
Reflection of a line segment in a point: Let O be the point in which we require to find the reflection of line segment PQ.
Steps of Construction:
1. Join PO and produce to P' such that PO = OP'.
2. Join QO and produce to Q, such that OO = OQ.
3. Join P' and Q'.
Then P'Q' is the reflection of line segment PQ in the point O.
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