Coordinates of the Image: (0,0), (0,60), (80,60), (80,0) Change: We need to change the unit value in the preimage to 3 feet.
Practice makes perfect
We are told that the location of a water treatment plant is mapped using a coordinate plane. The vertices of the plant are (0,0), (0,180), (240,180), and (240,0). Let's draw the plant by plotting the given points and connect them with segments.
We want to find the coordinates of the plant after we apply a dilation with a scale factor of 13. Before we do this, remember that 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 plant after a dilation with a scale factor k= 13.
Dilation With Scale Factor k= 13
Preimage
Multiply by k
Image
(0,0)
( 1/3(0), 1/3(0))
(0,0)
(0,180)
( 1/3(0), 1/3(180))
(0,60)
(240,180)
( 1/3(240), 1/3(180))
(80,60)
(240,0)
( 1/3(240), 1/3(0))
(80,0)
We can now plot the new points and connect them with segments to draw the image.
Notice that 1 unit of the preimage is equal to 3 units of the image. Since 1 unit of the preimage represents 1 foot, each unit of the image represents 13 feet.
To represent the location of the plant with the image, we will adjust each unit of the image to represent 1 foot. The only way this is possible is if after the dilation, each unit of the preimage represents 3 feet on the image. Therefore, we need to change the unit value in the preimage to 3 feet.
