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Transformations of Parallel and Perpendicular Lines

Concept

Creating Parallel and Perpendicular Lines through Rigid Transformations

When a line undergoes rigid transformations — namely a translation, rotation or reflection — it's possible for the preimage and the image to create a pair of parallel or perpendicular lines.

Concept

Parallel Lines

Parallel lines are lines that never intersect each other. Therefore, when creating parallel lines through rigid motions, it's necessary that the transformation doesn't change the direction of the line. In other words, the image should have the same slope as the preimage.

Therefore, a translation or a reflection are possible transformations to create parallel lines. If a reflection is used, the line of reflection has to be parallel with the given line. The last rigid motion, a rotation, can not be used to create parallel lines since it changes the direction of the line.

Concept

Perpendicular Lines

Perpendicular lines intersect each other at a right angle. Therefore, the transformation should create an image that intersects the preimage at an angle of

Since a rotation changes the angle between the preimage and the image, it can create perpendicular lines. If a reflection is used, the angle between the line and the line of reflection must be This will create a right angle between the preimage and the image. It is not possible to use a translation since it doesn't change the angle between the lines.
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Exercise

Given the line below, show two different rigid motions that creates a pair of parallel lines.

Show Solution
Solution

We must transform the given line so that the image created is parallel with the preimage. The lines will be parallel if they never intersect. If we look at a translation, it will move the line without changing its direction at all. Any translation will work, but here we will use a vertical translation.

The preimage and image are parallel since they have the same direction and will never intersect. Therefore, a translation will create a pair of parallel lines. Alternately, if we reflect the given line in a line of reflection parallel to it, the image will also be parallel.

The last rigid motion is rotation and it will rotate the line around a given point. Since this will change the direction of the line, it's not possible to use this transformation to create parallel lines.

fullscreen
Exercise

Given the line below, show two different rigid motions that create a pair of perpendicular lines.

Show Solution
Solution

Perpendiculars lines are lines that intersect at a right angle. Therefore, we should find rigid motions that create an image that intersects the given line at One rigid motion that changes the angle between the image and the preimage is a rotation. By rotating the line we will create a right angle between the lines.

Therefore, rotations can be used to create perpendicular lines. It is also possible to rotate the given line instead by  because the lines will still intersect at a right angle.

There are two more rigid motions, translation and reflection. A translation will only move the line, not rotate it. Therefore, it will not create perpendicular lines. A reflection could be used depending on the line of reflection.

If the angle between the preimage and the line of reflection is , the angle between the preimage and the image will be Similar to a rotation, if the angle between the preimage and the line of reflection is perpendicular lines will be created.

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