12.3: Reflections
Learning Objectives
 Reflect a figure over a given line.
 Determine the rules of reflections in the coordinate plane.
Review Queue
 Define reflection in your own words.
 Plot
A(−3,2) . TranslateA such that(x,y)→(x+6,y) .  What line is halfway between
A andA′ ?  Translate
A such that(x,y)→(x,y−4) . Call this pointA′′ .  What line is halfway between
A andA′′ ?
Know What? A lake can act like a mirror in nature. Describe the line of reflection in the photo below. If this image were on the coordinate plane, what could the equation of the line of reflection be? (There could be more than one correct answer, depending on where you place the origin.)
Reflections over an Axis
The next transformation is a reflection. Another way to describe a reflection is a “flip.”
Reflection: A transformation that turns a figure into its mirror image by flipping it over a line.
Line of Reflection: The line that a figure is reflected over.
Example 1: Reflect
Solution: To reflect
From this example, we can generalize a rule for reflecting a figure over the
Reflection over the
Example 2: Reflect the letter “
Solution: To reflect the letter
The generalized rule for reflecting a figure over the
Reflection over the
Reflections over Horizontal and Vertical Lines
Other than the
Example 3: Reflect the triangle
Solution: Notice that this vertical line is through our preimage. Therefore, the image’s vertices are the same distance away from
Example 4: Reflect the line segment
Solution: Here, the line of reflection is on
Reflection over
Reflection over
From these examples we also learned that if a point is on the line of reflection then the image is the same as the original point.
Example 5: A triangle
Solution: Looking at the graph, we see that the preimage and image intersect when
If the image does not intersect the preimage, find the midpoint between a preimage and its image. This point is on the line of reflection. You will need to determine if the line is vertical or horizontal.
Reflections over
Technically, any line can be a line of reflection. We are going to study two more cases of reflections, reflecting over
Example 6: Reflect square
Solution: The purple line is
From this example, we see that the
Reflection over
Example 7: Reflect the trapezoid TRAP over the line
Solution: The purple line is
From this example, we see that the
Reflection over
At first glance, it does not look like
From all of these examples, we notice that a reflection is an isometry.
Know What? Revisited The white line in the picture is the line of reflection. This line coincides with the water’s edge. If we were to place this picture on the coordinate plane, the line of reflection would be any horizontal line. One example could be the
Review Questions
 Which letter is a reflection over a vertical line of the letter “
b ”?  Which letter is a reflection over a horizontal line of the letter “
b ”?
Reflect each shape over the given line.

y− axis 
x− axis 
y=3 
x=−1 
x− axis 
y− axis 
y=x 
y=−x 
x=2 
y=−4 
y=−x 
y=x
Find the line of reflection of the blue triangle (preimage) and the red triangle (image).
Two Reflections The vertices of
 Plot
△ABC on the coordinate plane.  Reflect
△ABC overy=1 . Find the coordinates of△A′B′C′ .  Reflect
△A′B′C′ overy=−3 . Find the coordinates of△A′′B′′C′′ .  What one transformation would be the same as this double reflection?
Two Reflections The vertices of
 Plot
△DEF on the coordinate plane.  Reflect
△DEF overx=2 . Find the coordinates of△D′E′F′ .  Reflect
△D′E′F′ overx=−4 . Find the coordinates of△D′′E′′F′′ .  What one transformation would be the same as this double reflection?
Two Reflections The vertices of
 Plot
△GHI on the coordinate plane.  Reflect
△GHI over thex− axis. Find the coordinates of△G′H′I′ .  Reflect
△G′H′I′ over they− axis. Find the coordinates of△G′′H′′I′′ .  What one transformation would be the same as this double reflection?
 Following the steps to reflect a triangle using a compass and straightedge.
 Make a triangle on a piece of paper. Label the vertices
A,B andC .  Make a line next to your triangle (this will be your line of reflection).
 Construct perpendiculars from each vertex of your triangle through the line of reflection.
 Use your compass to mark off points on the other side of the line that are the same distance from the line as the original
A,B andC . Label the pointsA′,B′ andC′ .  Connect the new points to make the image
△A′B′C′ .
 Make a triangle on a piece of paper. Label the vertices
 Describe the relationship between the line of reflection and the segments connecting the preimage and image points.
 Repeat the steps from problem 28 with a line of reflection that passes through the triangle.
Review Queue Answers
 Examples are: To flip an image over a line; A mirror image.

A′(3,2)  the
y− axis 
A′′(−3,−2)  the
x− axis
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Date Created:
Feb 23, 2012Last Modified:
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