Reflections
Another transformation that still maintains its size and shape is called a reflection. A reflection is an isometry that creates a mirror image of itself across a line called a line of reflection or line of symmetry. The pre-image and image are congruent figures. Take a look at the following examples of reflections and how we denote the line of reflection:
Below is a reflection of figure #1 in the y-axis. Below is a reflection of figure #1 in the x-axis
Another transformation that still maintains its size and shape is called a reflection. A reflection is an isometry that creates a mirror image of itself across a line called a line of reflection or line of symmetry. The pre-image and image are congruent figures. Take a look at the following examples of reflections and how we denote the line of reflection:
Below is a reflection of figure #1 in the y-axis. Below is a reflection of figure #1 in the x-axis
When we have specific coordinates where the figures are placed, we can be more precise about finding the coordinates of the image of a reflection.
Example 1: Let’s reflect a figure in a vertical line:
Find \(r_{x = 1}\). In other words, reflect the following triangle in the line \(x = 1\). Recall that \(x = 1\) is a vertical line. This is really important, since a reflection in a horizontal line would yield a different image!
Example 1: Let’s reflect a figure in a vertical line:
Find \(r_{x = 1}\). In other words, reflect the following triangle in the line \(x = 1\). Recall that \(x = 1\) is a vertical line. This is really important, since a reflection in a horizontal line would yield a different image!
Notice a few properties that occur with reflections:
Example 2:
Let’s reflect a figure in a horizontal line. Find \(r_{y=2}\). In other words, reflect the following figure in the line \(y = 2\). Recall that the distance from pre-image to the line of reflection should be the same as the distance from line of reflection to its image. The distances in this reflection must be counted vertically since the line of reflection is horizontal!
- The perpendicular distance from point C to the line of reflection is the same as the distance from point C’. This should always happen between image and pre-image point. Check the distances for points A and A’ too!
- Notice, since point B is on the line of reflection, its image is the same point. So the coordinates of point B and point B’ are the same.
Example 2:
Let’s reflect a figure in a horizontal line. Find \(r_{y=2}\). In other words, reflect the following figure in the line \(y = 2\). Recall that the distance from pre-image to the line of reflection should be the same as the distance from line of reflection to its image. The distances in this reflection must be counted vertically since the line of reflection is horizontal!
When the line of reflection is neither vertical nor horizontal, we will still find reflections the same way: the distance from the pre-image to line of reflection must be the same as the distance to the image.
Example 3:
Reflect a figure in a non-vertical or a non-horizontal line. Find \(r_{y=x}\). In other words, reflect the figure in the line \(y = x\). List out the coordinates of its image.
Example 3:
Reflect a figure in a non-vertical or a non-horizontal line. Find \(r_{y=x}\). In other words, reflect the figure in the line \(y = x\). List out the coordinates of its image.
Solution:
Let’s watch this video to find its solution.
Let’s watch this video to find its solution.
Properties of reflections:
- The line of reflection must be the perpendicular bisector of the segment joining the pre-image to the image.
- The pre-image and image must be the same shape and size, which means a reflection is an isometry!
Reflections Rules: If a point \((x, y)\) is reflected in the following lines, here is a summary of how to find its image.
Coordinates of Pre-image |
Line of reflection |
Coordinates of Image |
\((x, y)\) |
\(x\)-axis |
\((x, -y)\) |
\((x, y)\) |
\(y\)-axis |
\((-x, y)\) |
\((x, y)\) |
\(y = x\) |
\((y, x)\) |
\((x, y)\) |
\(y = -x\) |
\((-y, -x)\) |
Example 4:
Reflect the graph in the \(x\)-axis.
Reflect the graph in the \(x\)-axis.
Solution:
Reflecting in the \(x\)-axis transforms the coordinates from \((x, y)\) to \((x, -y)\). The points on the graph such as \((2, 1), (0, 5),\) and \(4, 5)\) will reflect to \(2, -1), (0, -5)\) and \((4, -5)\).
Reflecting in the \(x\)-axis transforms the coordinates from \((x, y)\) to \((x, -y)\). The points on the graph such as \((2, 1), (0, 5),\) and \(4, 5)\) will reflect to \(2, -1), (0, -5)\) and \((4, -5)\).
If you reflect in any lines that are different from the ones listed in the table, it will be easiest to use the properties of reflections to help you find the reflection image.
FINDING MINIMUM DISTANCE
We can use reflections to help us find a minimum distance between two points! Let’s take a look at the example below.
Example 5:
Given \(A (0, 3)\) and \(B (8, 1)\), find point \(C\) on the x-axis that will give the minimum distance between points \(A\) and \(B\). In other words, put point \(C\) on the \(x\)-axis that will make \(AC + CB\) the shortest distance!
FINDING MINIMUM DISTANCE
We can use reflections to help us find a minimum distance between two points! Let’s take a look at the example below.
Example 5:
Given \(A (0, 3)\) and \(B (8, 1)\), find point \(C\) on the x-axis that will give the minimum distance between points \(A\) and \(B\). In other words, put point \(C\) on the \(x\)-axis that will make \(AC + CB\) the shortest distance!
Solution:
Step 1: Reflect points \(A\) and \(B\) into the \(x\)-axis:
Step 1: Reflect points \(A\) and \(B\) into the \(x\)-axis:
Step 2: Connect \(\overline{AB’}\) with a straightedge. Where it meets the \(x\)-axis is your point \(C\)
Just for good measure, you can also connect \(\overline{A’B}\). It should meet the x-axis at the same point:
So the point on the \(x\)-axis that makes the shortest distance is (\(6, 0)\). If you measure the distance from \(A\) to \(C\), and add the distance from \(C\) to \(B\), it will be the shortest distance than if you choose a different point on the \(x\)-axis!
So the point on the \(x\)-axis that makes the shortest distance is (\(6, 0)\). If you measure the distance from \(A\) to \(C\), and add the distance from \(C\) to \(B\), it will be the shortest distance than if you choose a different point on the \(x\)-axis!