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Explore

Reflecting a Triangle

In the following applet, the vertices of can be moved. Also, the slope of the line can be set by moving the slider point. Once everything is set, reflect across line
Is there any relationship between and If so, do and have the same relationship? What about and

Discussion

Properties of a Reflection

As can be checked in the previous exploration, after a point is reflected across a line the segment connecting with its image is perpendicular to Additionally, the line intersects at its midpoint.
A point and its reflection across a line
From these two facts, it can be concluded that is the perpendicular bisector of With this information in mind, reflections can be defined properly.

Discussion

Reflections

Example

Reflecting a Polygon

The principal of Jefferson High wants to build a physics lab by the chemistry lab. The plan, seen from the sky, is that the new building looks like a reflection of the chemistry lab through the walkway that connects the soccer field with the library.

A quadrilateral and a line

Perform a reflection to the chemistry lab across the walkway in order to draw the physics lab.

Answer
Quadrilateral ABCD and its image under the reflection across ell

Hint

To reflect the chemistry lab, reflect each corner of the building. The physics lab is the quadrilateral formed by the images. Remember that the image of a point that is on the line of reflection is the same point.

Solution

For simplicity, start by labeling each corner of the quadrilateral and the walkway.

Quadrilateral ABCD and line ell

To reflect across a reflection can be performed on each vertex, one at a time. For example, to reflect is a good start. To do so, follow the definition of reflections. First, draw a line perpendicular to passing through

Line perpendicular to ell passing through A

Then can be plotted as the point on line where its distance to is the same as the distance from to

Image of A over the line m

The same steps can be applied to reflect vertices and Notice that because both and are on the line their images will maintain their same point locations, respectively.

Image of A, B, C, and D

Finally, the image of under a reflection across the line is the quadrilateral formed by and This quadrilateral represents the physics lab.

Quadrilateral ABCD and its image under the reflection across ell

Discussion

Reflections Performed by Hand

Reflections can be performed by hand with the help of a straightedge and a compass.

Triangle ABC and line ell

To reflect across the line follow these three steps.

1
Draw an Arc Centered at that Intersects at Two Points
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Place the compass tip on vertex and draw an arc that intersects the line at two different points. Let and be these points.
Drawing an Arc centered at A that intersects ell at two points
2
Draw Two Arcs Centered at the Intersection Points
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With the same compass setting, place the compass tip on and draw an arc. Keeping the same setting, place the compass tip on and draw a second arc. The intersection point of these two arcs is the image of
Drawing arcs centered at P and Q that intersect each other
Notice that both arcs need to be drawn on the side of not containing vertex
3
Repeat the Previous Steps for the Other Vertices
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To reflect and repeat the two previous steps.

Images of A, B, and C

The image of after the reflection is the triangle formed by and

Triangle ABC and triangle A'B'C'
Notice that the first two steps are the same as the first two steps to construct a line perpendicular to a given line through a point.

Discussion

Reflections in the Coordinate Plane

In the coordinate plane, there is a particular relationship between the coordinates of a point and those of its image after a reflection across certain lines worth considering. These lines are the coordinate axes and lines and Investigate each relationship by using the following applet.
Applet to investigate the coordinates of a point after a reflection across the coordinate axes and the lines y=x and y=-x
Drawn from diagram, the following relations can be determined.
  • The image of under a reflection across the axis is
  • The image of under a reflection across the axis is
  • The image of under a reflection across the line is
  • The image of under a reflection across the line is

Pop Quiz

Practicing Reflections

In the following applet, there are two possible requests.

To reflect place points and where they should be after the reflection is applied. To draw the line of reflection, place the two points, so they lie on the line of reflection.

Performing random reflections to random triangles

Discussion

Reflections Across Parallel Lines

In the previous example, it was concluded that the composition of reflections in parallel lines gives the same result as a translation. This conclusion is not an isolated fact. Actually, there is a theorem that guarantees this result.

Discussion

Reflections Across Intersecting Lines

Additionally, there is also a theorem for the case where the lines of reflection intersect each other.

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