Point $V$ is somewhere on $\overline{U W}$. If $U V=2 x-13$, $V W=-18+2 x$, and $U W=17$, then find the length of $\overline{U V}$.
Given that point \(V\) lies on the segment \(\overline{UW}\), we apply the Segment Addition Postulate, which states that the sum of the lengths of the two smaller segments equals the length of the entire segment:
\[UV + VW = UW\]
We are given the expressions for the lengths of these segments:
\(UV = 2x - 13\)
\(VW = -18 + 2x\)
\(UW = 17\)
By substituting these expressions into the postulate, we obtain the following linear equation:
\[(2x - 13) + (-18 + 2x) = 17\]
To solve for \(x\), we first combine the like terms on the left side of the equation:
\((2x + 2x) + (-13 - 18) = 17\)
\(4x - 31 = 17\)
Now, add \(31\) to both sides of the equation:
\(4x = 17 + 31\)
\(4x = 48\)
Dividing both sides by \(4\), we find the value of \(x\):
\(x = \frac{48}{4} = 12\)
Now that we have determined the value of \(x = 12\), we substitute it back into the expression for the length of segment \(\overline{UV}\):
\(UV = 2x - 13\)
\(UV = 2(12) - 13\)
\(UV = 24 - 13\)
\(UV = 11\)
To verify our answer, we can also calculate the length of \(\overline{VW}\):
\(VW = -18 + 2(12) = -18 + 24 = 6\)
Since \(UV + VW = 11 + 6 = 17\), which matches the given total length \(UW = 17\), our result is confirmed.
The correct option is a. 11.
The Segment Addition Postulate is frequently used in construction and navigation. For instance, if you are measuring a distance along a straight path between two landmarks, say Landmark A and Landmark C, and you stop at Landmark B along the way, the total distance from A to C is simply the distance from A to B plus the distance from B to C. By knowing these individual segments, you can verify your total trip distance and ensure your measurements are accurate.