Exercise 3 Page 677

We will begin by determining whether the given dilation from figure $B$ to $B^{\prime }$ 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 $\text{larger}$ than the preimage, the dilation was an $\text{enlargement}.$ If the image is $\text{smaller}$ than the preimage, the dilation was a $\text{reduction}.$

We can see that the image $B^{\prime }$ 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 $Q{B}^{\prime }$ is equal to the scale factor $k$ multiplied by $QB.$ Knowing the lengths of $QB$ and $Q{B}^{\prime }$ we can substitute them in the above equation and solve for $k.$ The scale factor of the given dilation is $\frac{4}{3}.$

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. Also, notice that the point $B$ lies on the same line as the points $Q$ and $B^{\prime }.$ Therefore, we can express the length $Q{B}^{\prime }$ as the sum of $x$ and $6.$ Because we are given the length of $Q{B}^{\prime },$ we can substitute it into the equation to find $x.$