Reflected points are the same distance from but on opposite sides of the line of reflection before and after the reflection takes place.
Practice makes perfect
We are given the vertices of triangle FGH. We will start by graphing this points and connect them to form â–ł FGH.
We want to find the image of this triangle after a similarity transformation. Let's do one transformation at time!
Reflection
To reflect this figure over the x-axis, we need to plot each vertex of the image F'G'H' the same distance from the line of reflection as its vertex on the preimage FGH. 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 FGH
Image F'G'H'
Vertex
Distance From the x-axis
Vertex
Distance From the x-axis
F(1,2)
2 units above the x-axis
F''(1, - 2)
2 units below the x-axis
G(4,4)
4 units above the x-axis
G''(4,- 4)
4 units below the x-axis
H(2,0)
0 units below the x-axis
H''(2,0)
0 units above the x-axis
Now that we know the coordinates of â–ł F'G'H' we can draw its image after the similarity transformation.
Dilation
A dilation can be an enlargement, a reduction, or the same size as the preimage. Which type of dilation it is depends on the value of the scale factor k.
Enlargement
k>1
Reduction
0
Same
k=1
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 FGH after a dilation with a scale factor k= 1.5.
Dilation With Scale Factor k=1.5
Preimage
Multiply by k
Image
F'(1,- 2)
( 1.5 (1), 1.5 (- 2))
F''(1.5,- 3 )
G'(4,- 4)
( 1.5 (4), 1.5 (- 4))
G''(6,- 6)
H'(2,0)
( 1.5 (2), 1.5 (0))
H''(3, 0)
We can now plot the obtained points and connect them with segments to draw the image.
Finally, we can remove the intermediate step and have our transformation.
