A rotation by 180^(∘) changes point (x,y) into point (- x, - y).
B
Reflecting a point across the x-axis changes the y-coordinate into its opposite, while the x-coordinate stays unchanged.
A
A'(6, 2), B'(3, 2), C'(1, 4), D'(6, 4)
B
Example Answer: Reflect the trapezoid across the x-axis and then across the y-axis.
Practice makes perfect
The given trapezoid has the following vertices.
A(-6, -2), B(-3, -2), C(-1, -4), D(-6, -4)
We want to find the coordinates of the image of this trapezoid after the 180^(∘) rotation. This rotation changes both the x- and y-coordinates into their opposites.
( x, y)
⟶
( - x, - y)
Let's apply this transformation to the vertices of the given trapezoid.
(x,y)
(- x, - y)
Simplify
A( -6, -2)
A'( -( -6) , -( -2) )
A'(6, 2)
B( -3, -2)
B'( -( -3) , -( -2) )
B'(3, 2)
C( -1, -4)
C'( -( -1) , -( -4) )
C'(1, 4)
D( -6, -4)
D'( -( -6) , -( -4) )
D'(6, 4)
The rotated trapezoid has the following coordinates.
A'(6, 2), B'(3, 2), C'(1, 4), D'(6, 4)
To get these coordinates, we can also draw the original triangle and then rotate it. Let's take a look!
In Part A we found that the image of the given trapezoid in 180^(∘) rotation has the following coordinates.
A'(6, 2), B'(3, 2), C'(1, 4), D'(6, 4)
We want to get this image without using rotations. Let's try to do this using other transformations. First, let's remember how the rotation changed the coordinates of points.
( x, y)
⟶
( - x, - y)
Now we want to change both coordinates to their opposites without using rotations. Reflections across the axes change the sign of coordinates, but only one at a time. Let's recall how they work.
Transformation
Effect on Coordinates
Reflection in the x-axis
Changes the y-coordinate to its opposite
Reflection in the y-axis
Changes the x-coordinate to its opposite
If we perform only one of these transformations, we will not get to the desired result. However, we will if we perform both of them one after the other!
cc
( x, y) &
↓ & Reflection in thex-axis
( x, - y) &
↓ & Reflection in they-axis
( - x, - y) &
Performing two reflections leaves us with the same result as performing one 180^(∘) rotation about the origin. Since this works for any point, it will also work for the vertices our trapezoid!
