We will discuss here about reflection of a point in the xaxis.
Reflection in the line y = 0 i.e., in the xaxis.
The line y = 0 means the xaxis.
Let P be a point whose coordinates are (x, y).
Let the image of P be P’ in the axis.
Clearly, P’ will be similarly situated on that side of OX which is opposite to P. So, the ycoordinates of P’ will be – y while its xcoordinates will remain same as that of P.
The image of the point (x, y) in the xaxis is the point (x, y).
Symbolically, M\(_{x}\) (x, y) = (x, y)
Rules to find the reflection of a point in xaxis:
(i) Retain the abscissa i.e. xcoordinate.
(ii) Change the sign of ordinate i.e., ycoordinate.
Therefore, when a point is reflected in the xaxis, the sign of its ordinate changes.
Examples:
(i) The image of the point (3, 4) in the xaxis is the point (3, 4).
(ii) The image of the point (3, 4) in the xaxis is the point (3, (4)) i.e., (3, 4).
(iii) The reflection of the point (5, 7) in the xaxis = (5, 7) i.e., M\(_{x}\) (5, 7) = (5, 7)
(iv) The reflection of the point (9, 0) in the xaxis is the point itself, therefore, the point (9, 0) is invariant with respect to xaxis.
(v) The reflection of the point (a, b) in the xaxis = (a, b) i.e., M\(_{x}\) (a, b) = (a, b)
Solved examples to find the reflection of a point in the xaxis:
1. Find the points onto which the points (11, 8), (6, 2) and (0, 4) are mapped when reflected in the xaxis.
Solution:
We know that a point (x, y) maps onto (x, y) when reflected in the xaxis. So, (11, 8) maps onto (11, 8); (6, 2) maps onto (6, 2) and (0, 4) maps onto (0, 4).
2. Which of the following points (2, 0), (0, 5), (3, 3) are invariant points when reflected in the xaxis?
Solution:
We know that only those points which lie on the line are invariant points when reflected in the line. So, only those points are invariant which lie on the xaxis. Hence, the invariant points must have ycoordinate = 0. Therefore, only (2, 0) is the invariant point.
3. Which of the following points (7, 0), (1, 1), (2, 2), (0, 4) are invariant points when reflected in the xaxis?
Solution:
We know that only those points which lie on the line are invariant points when reflected in the line. So, only those points are invariant which lie on the yaxis. Hence, the invariant points must have xcoordinate = 0. Therefore, only (0, 4) is the invariant point.
10th Grade Math
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