Let's start by plotting the given points and connecting them with segments to draw the square.
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 square after a dilation with a scale factor k= 0.5.
Dilation With Scale Factor k=0.5
Preimage
Multiply by k
Image
(0,0)
( 0.5(0), 0.5(0))
(0,0)
(0,4)
( 0.5(0), 0.5(4))
(0,2)
(4,4)
( 0.5(4), 0.5(4))
(2,2)
(4,0)
( 0.5(4), 0.5(0))
(2,0)
Now we can plot the new points on the coordinate plane and connect them with segments to draw the image. Let's do it!
We want to find the ratio of the perimeters of the two figures. Remember that when two figures are similar, the value of the ratio of their perimeters is equal to the value of the ratio of their corresponding side lengths.
Perimeter of Image/Perimeter of Preimage=Side Length Image/Side Length Preimage
In our case, one of the side lengths of the original square is 4 and its corresponding side length is 2. Let's substitute these values into the above equation to find the ratio between the perimeters.
Perimeter of Image/Perimeter of Preimage=Side Length Preimage/Side Length Image
The ratio of the perimeter of the image to the perimeter of the preimage is 12. Now we can find the ratio of the areas. To do so, recall that when two figures are similar, the value of the ratio of their areas is equal to the square of the value of the ratio of their corresponding sides lengths.
Area of Image/Area of Preimage=(Side Length Image/Side Length Preimage)^2
The length of one of the side of the original square is 4 and its corresponding side length in the image is 2. Let's substitute these values into the equation to find the ratio of the areas.
Area of Image/Area of Preimage=(Side Length Image/Side Length Preimage)^2
