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?Enlargement Scale Factor: 4/3 Value of x: 2
Practice makes perfect
We will begin by determining whether the given dilation from figure B to B' is an enlargement or a reduction. Then we can find the value of the scale factor of the dilation and x.
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 B' is larger than than the preimage B, so this dilation is an enlargement.
Finding k and x
Now, let's find the value of the scale factor k of the dilation and x.
Scale Factor k
Notice that the point Q is the center of dilation in the diagram. By the definition of a dilation, the length QB' is equal to the scale factor k multiplied by QB.
QB'=k(QB) ⇔ k =QB'/QB
Knowing the lengths of QB and QB' we can substitute them in the above equation and solve for k.
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.
QB'=8, BB'=x, and QB=6
Also, notice that the point B lies on the same line as the points Q and B'.
Therefore, we can express the length QB' as the sum of x and 6.
QB'=x+6
Because we are given the length of QB', we can substitute it into the equation to find x.
