Reflected points are the same distance from, but on opposite sides of, the line of reflection before and after the reflection takes place.
Yes, see solution.
Practice makes perfect
We are told that the red figure is similar to the blue figure.
We want to know if we can get the blue figure after a reflection followed by a dilation of the red figure. Let's apply one transformation at time!
Reflection
To reflect the red figure over the x-axis, we need to plot each vertex of the image the same distance from the line of reflection as its corresponding vertex on the preimage. Because our line of reflection is the x-axis, this will change the sign of the y-coordinates of the points, but the x-coordinates will remain unchanged.
Preimage
Image
Vertex
Distance From the x-axis
Vertex
Distance From the x-axis
(4,- 2)
2 units below the x-axis
(4,2)
2 units above the x-axis
(8,- 2)
2 units below the x-axis
(8, 2)
2 units above the x-axis
(6,- 6)
6 units below the x-axis
(6, 6)
6 units above the x-axis
(4,- 4)
4 units below the x-axis
(4,4)
4 units above the x-axis
Let's do the reflection!
Dilation
A dilation can be an enlargement or a reduction of the preimage. Which type of dilation it is depends on the value of the scale factor k.
Enlargement
k>1
Reduction
0
When the center of dilation in the coordinate plane is the origin, each coordinate of the preimage is multiplied by the scale factor k to find the coordinates of the image.
ccc
Preimage & & Image [0.5em]
(x,y)& ⇒ & ( kx, ky)
Now, let's find the coordinates of the vertices of the red figure after a dilation with a scale factor k= 12.
Dilation With Scale Factor k= 12
Preimage
Multiply by k
Image
(4,2)
( 1/2(4), 1/2(2))
(2,1)
(8,2)
( 1/2(8), 1/2(2))
(4,1)
(6,6)
( 1/2(6), 1/2(6))
(3,3)
(4,4)
( 1/2(4), 1/2(4))
(2,2)
We can now draw the image after the dilation.
Therefore, we can get the blue figure with the given sequence of rigid motions. This is possible because if if two figures are similar, the final image will not depend on the order of transformations.
