what is a reflection?
Formal Definition: A reflection (or flip) is an isometry in which a figure and its image have opposite orientations.
Understandable Definition: A reflection is the flipping of a shape without translating (sliding) or rotating it AND keeping it the same size.
Understandable Definition: A reflection is the flipping of a shape without translating (sliding) or rotating it AND keeping it the same size.
When we are instructed to flip a figure, one of the questions that comes up is "Which way do we flip it?"
This question identifies something very important we have to know before we start reflecting our figures. This is called the line of reflection or axis of reflection.
In the picture below we see two examples of lines of reflection and what a figure looks like when it's flipped over these lines.
This question identifies something very important we have to know before we start reflecting our figures. This is called the line of reflection or axis of reflection.
In the picture below we see two examples of lines of reflection and what a figure looks like when it's flipped over these lines.
When our figures are flipped, both blue points end up the same distant from the line of reflection. The same thing happens for the red and green points. This is an easy way to check if we did our reflection correctly.
reflections in kangas
Reflections are used in kanga designs when symmetry is desired. The same shape can be flipped in a number of ways to vary the design without incorporating new figures. Here are some examples of reflections in kanga designs:
describing reflections
Given a geometric figure (preimage) and the transformed figure (image), we can describe the reflection. This works the best when our figures are graphed on a coordinate plane.
If we reflect our rhombus over the X axis, as shown to the right, the points of our reflected figure, A'B'C'D' are (1.5, -3), (4, -3), (3.5, -1), and (1, -1), respectively. We can compare the points of our preimage (rhombus ABCD) to our image (rhombus A'B'C'D') to see how we describe a vertical flip. A B C D (1.5, 3) (4, 3) (3.5, 1) (1, 1) (1.5, -3) (4, -3) (3.5, -1) (1, -1) This is one example of the how a vertical flip (reflection over the X axis) changes the Y coordinate of each point of our figure. Reflection Over X Axis: (x,y) to (x, -y)
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If we reflect our rhombus over the Y axis, as shown to the left, the points of our reflected figure, A'B'C'D' are (-1.5, 3), (-4, 3), (-3.5, 1), and (-1, 1), respectively. We can compare the points of our preimage (rhombus ABCD) to our image (rhombus A'B'C'D') to see how we describe a horizontal flip. A B C D (1.5, 3) (4, 3) (3.5, 1) (1, 1) (-1.5, 3) (-4, 3) (-3.5, 1) (-1, 1) This is one example of the how a horizontal flip (reflection over the Y axis) changes the X coordinate of each point of our figure. Reflection Over Y Axis: (x,y) to (-x, y)
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We can compare the points of our preimage (triangle PQR) to our image (triangle P'Q'R') to see how we describe a reflection over the blue line y = x. P Q R (1, -1) (4, -3) (1, -3) (-1, 1) (-3, 4) (-3, 1) This is one example of the how a reflection over the line y = x "flip-flops" the X and Y coordinates of each point of our figure. Reflection Over y = x: (x,y) to (y,x)
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fascinating reflections
Some figures can be reflected onto themselves depending on the line of reflection. Some examples are shown below.
This kanga design is also reflected directly onto itself when it's reflected over both the X axis and Y axis.
The beauty of reflections found in nature is best demonstrated by butterflies. Their bodies and, more importantly, their wings show us perfect examples of this.
try it out!
As we worked our way through this webpage, we attempted to master the underlined parts of the following Common Core State Standards:
- Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
- Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
- Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
We also attempted to master the following Tanzania National Standards:
- The student should be able to state properties of reflections.
- The student should be able to represent reflections by drawing.
- Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
- Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
- Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
We also attempted to master the following Tanzania National Standards:
- The student should be able to state properties of reflections.
- The student should be able to represent reflections by drawing.
Let's see how we did!
Click on the following two buttons to demonstrate mastery of these skills. The first webpage will ask you to construct reflections and the second will ask you to find the coordinates of the reflections.
Click on the following two buttons to demonstrate mastery of these skills. The first webpage will ask you to construct reflections and the second will ask you to find the coordinates of the reflections.