In a parallelogram, the diagonals bisect each other. Therefore, in parallelogram ABCD, point O, being the intersection of the diagonals AC and BD, indicates that AO = OC and BO = OD.
Since you mentioned that \( AC = 48 \), point O being the midpoint means that:
\[ AO = OC = \frac{AC}{2} = \frac{48}{2} = 24 \]
Now we also know that \( DO = 7x + 3 \) and using the fact that diagonals bisect each other, we have:
\[ BO = OD \]
Because AO and DO refer to the same segments from O to the corners, we apply the equality:
\[ AO = DO \] \[ 24 = 7x + 3 \]
To solve for \( x \), we will subtract 3 from both sides:
\[ 24 - 3 = 7x \] \[ 21 = 7x \]
Next, divide both sides by 7:
\[ x = 3 \]
Thus, the value of \( x \) that proves that parallelogram ABCD is a rectangle is:
\[ \boxed{3} \]