If the image is larger than the preimage, the dilation was an enlargement and the scale factor will be greater than 1. If the image is smaller than the preimage, the dilation was a reduction and the scale factor will be less than 1.
Enlargement or Reduction?Reduction Scale Factor: 1/3 Value of x: 4
Practice makes perfect
We will begin by determining whether the given dilation from figure W to W' is an enlargement or a reduction. Then we can find the value of x and the scale factor of the dilation.
Type of Dilation
If the image is larger than the preimage, the dilation was an enlargement. If the image is smaller than the preimage, the dilation was a reduction.
We can see that the image W' is smaller than than the preimage W, so this dilation is a reduction.
Finding x and k
Now, let's find the value of x and the scale factor k of the dilation.
Value of x
In order to find the value of x, we should consider all of our known information. On the diagram, we are given the lengths of a few segments.
FW=12, FW'=x, and WW'=8
Also, notice that the point W' lies on the same line as the points F and W.
Therefore, we can express the length FW as the sum of x and 8.
FW=x+8
Because we are given the length of FW, we can substitute it into the equation to find x.
Next, notice that the point F is the center of dilation in the diagram. By the definition of a dilation, the length FW' is equal to the scale factor k multiplied by FW.
FW'=k(FW) ⇔ k =FW'/FW
Knowing the value of x, we can substitute it in FW'=x and conclude that the length FW' is equal to 4.
FW'=4
FW=12
Finally, we can substitute these values in our expression for the scale factor k and simplify.
