Exercise 14 Page 329

We need to find the measure of $SW.$ Let's consider the given diagram.

As we can see, segment $\overline{SW}$ is the sum of the segments $\overline{ST}$ and $\overline{TW}.$ By the Segment Addition Postulate, its length equals the sum of these segments' lengths. From the diagram, we know that $2x+2$ represents the length of $\overline{ST},$ and $4x-4$ represents the length of $\overline{TW}.$ Let's substitute these into the above equation, and find the expression for the length of $\overline{SW}.$ Thus, to find $SM,$ we need to know the value of $x.$ How can we calculate it?

From the diagram, we can see that distances $RW$ and $RS$ are the same. This means that point $R$ is equidistant from the endpoints $S$ and $W$ of the segment $\overline{SW}.$

Now, we can use the Converse of the Perpendicular Bisector Theorem. According to the theorem, $R$ lies on the perpendicular bisector of the segment $\overline{SW}.$ Thus, $RT,$ which is perpendicular to $SW,$ is also a bisector of $SW.$ Therefore, $ST=TW.$ Substituting $2x+2$ for $ST$ and $4x-4$ for $TW,$ we can write the following equation. Let's solve it and find the value of $x.$ Now that we know the value of $x,$ we can calculate the measure of $SW.$ The measure of $SW$ is $16.$