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Reflection over the line is a geometric transformation that maps every point of an object to a new position with the line acting as a mirror.
The distance from the original point to the mirror line is equal to the distance from the image point to the mirror line, and the line connecting the original point to its image will be perpendicular to the line .
If a point is reflected over the line , its image's coordinates, , will follow a simple rule:
Thus, the image of point is . Notice that the and coordinates swap positions.
Suppose we have point . If point is reflected over the line , its image, , can be determined by swapping its coordinates:
The original -coordinate is , becoming the new -coordinate.
The original -coordinate is , becoming the new -coordinate.
Thus, the image of point is .
Let's visualize this along with some other points:
Determine the image of triangle with vertices , , and reflected over the line .
To reflect the triangle, we reflect each of its vertices over the line :
The image triangle is formed by connecting the points , , and .
If a line has the equation is reflected over the line , determine the equation of its image.
To find the equation of the image, we use the rule and . This means we replace every in the original equation with (or ) and every with (or ).
Original equation:
Substitute and (using and for the new variables for simplicity):
This is the equation of the image line. Usually, we rewrite this equation in the form as a function of :
So, the equation of the image of the line after reflection over is .
The image of point is .
Explanation: The and coordinates are swapped: , .
The coordinates of the image triangle are:
The equation of the image of the line is .
Explanation: Substitute and into the original equation:
If converted to the form :
