Certainly! Let’s solve the problem step-by-step.
We know the following information about the points [tex]\( U \)[/tex], [tex]\( V \)[/tex], and [tex]\( W \)[/tex]:
- [tex]\( UV = 2x - 13 \)[/tex]
- [tex]\( VW = -18 + 2x \)[/tex]
- [tex]\( UW = 17 \)[/tex]
Since [tex]\( V \)[/tex] is a point on [tex]\(\overline{UW}\)[/tex], we know that:
[tex]\[ UV + VW = UW \][/tex]
Substitute the given expressions into the equation:
[tex]\[ (2x - 13) + (-18 + 2x) = 17 \][/tex]
Combine like terms:
[tex]\[ 2x - 13 - 18 + 2x = 17 \][/tex]
[tex]\[ 4x - 31 = 17 \][/tex]
Next, solve for [tex]\( x \)[/tex]:
[tex]\[ 4x - 31 = 17 \][/tex]
Add 31 to both sides:
[tex]\[ 4x = 48 \][/tex]
Divide by 4:
[tex]\[ x = 12 \][/tex]
Now that we have the value of [tex]\( x \)[/tex], we can find the length of [tex]\( UV \)[/tex]:
[tex]\[ UV = 2x - 13 \][/tex]
Substitute [tex]\( x = 12 \)[/tex]:
[tex]\[ UV = 2(12) - 13 \][/tex]
[tex]\[ UV = 24 - 13 \][/tex]
[tex]\[ UV = 11 \][/tex]
Therefore, the length of [tex]\(\overline{UV}\)[/tex] is [tex]\( 11 \)[/tex].
