A dilation can be an enlargement, a reduction, or the same size as the original figure. 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 original figure is multiplied by the scale factor k to find the coordinates of the image.
ccc
Preimage & & Image [0.5em]
(x,y)& ⇒ & ( kx, ky)
Let's find the coordinates of the vertices of rectangle QRST after a dilation with a scale factor k= 32.
Dilation With Scale Factor k= 32
Preimage
Multiply by k
Image
Q(- 6,2)
( 3/2(- 6), 3/2(2))
Q'(- 18/2,6/2)=(- 9,3)
R(6,2)
( 3/2(6), 3/2(2))
R'(18/2,6/2)=(9,3)
S(6,- 4)
( 3/2(6), 3/2(- 4))
S'(18/2,- 12/2)=(9,- 6)
T(- 6,- 4)
( 3/2(- 6), 3/2(- 4))
T'(- 18/2,- 12/2)=(- 9, - 6)
We can now plot the obtained points and connect them with segments to draw the image.
Translation
We want to draw the image of the obtained rectangle after a translation of 5 units to the right and 1 unit down. Therefore, we will add 5 to the x-coordinate and subtract 1 from the y-coordinate of each vertex.
Vertices of QRST
(x+5,y-1)
Vertices of Q'R'S'T'
Q'(- 9,3)
(- 9 + 5,3 - 1)
Q''(- 4,2)
R'(9,3)
(9 + 5,3 - 1)
R''(14,2)
S'(9,- 6 )
(9 + 5,- 6 - 1)
S''(14,- 7)
T'(- 9,- 6)
(- 9 + 5,- 6 - 1)
T''(- 4,- 7)
Let's do the translation!
The coordinates of the final image are Q''(- 4,2), R''(14,2), S''(14, - 7), and T''(- 4, - 7).
