Given the information that \( TURS \) is a rhombus, we know that the diagonals of a rhombus bisect each other at right angles. Therefore, the segments \( RV \) and \( VT \) are equal because they are both halves of diagonal \( RT \).
We have the expressions for \( RV \) and \( VT \): \[ RV = 2x + 4 \] \[ VT = 8x - 8 \]
Since \( RV = VT \), we can set the two expressions equal to each other: \[ 2x + 4 = 8x - 8 \]
Now, let's solve for \( x \):
- Subtract \( 2x \) from both sides: \[ 4 = 6x - 8 \]
- Add \( 8 \) to both sides: \[ 12 = 6x \]
- Divide both sides by \( 6 \): \[ x = 2 \]
Now that we have the value of \( x \), we can find \( VT \) by substituting \( x = 2 \) back into the expression for \( VT \): \[ VT = 8x - 8 \] \[ VT = 8(2) - 8 \] \[ VT = 16 - 8 \] \[ VT = 8 \]
Thus, the length of \( VT \) is \( \boxed{8} \).