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If a point is reflected over the line , its image's coordinates, , will follow a simple rule:
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.
The image of point is .
In the following point pairs, the coordinates exchange places:
Check the midpoint of the pair: it must lie on the mirror line. The joining segment must also be perpendicular to that line, giving the two points equal distances from it.
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 .
The line is reflected across . 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 .
Apply the same coordinate rule to the single point and to every triangle vertex. In the final problem, apply the rule to the line equation to obtain its entire set of image points.
Check the coordinate order when swapping coordinates. Each sign must follow the chosen reflection rule. For a line, simplify the equation after completing every substitution.
Solution 1
The image of point is .
Explanation: The and coordinates are swapped: , .
Solution 2
The coordinates of the image triangle are:
Solution 3
The equation of the image of the line is .
Explanation: Substitute and into the original equation:
If converted to the form :
Published: . Updated: .
