In a parallelogram, the diagonals bisect each other. This means that the lengths of the segments created by the intersection of the diagonals are equal.
Given that diagonal AC = 48, the length of each segment from the intersection point O to the endpoints of the diagonal AC (which is bisected at point O) is:
\[ AO = OC = \frac{AC}{2} = \frac{48}{2} = 24 \]
We also have that \(DO = 7x + 3\). Since AO and DO are equal in a parallelogram with congruent diagonals, we can set up the equation:
\[ 7x + 3 = 24 \]
Now, we will solve for \(x\):
- Subtract 3 from both sides:
\[ 7x = 24 - 3 \] \[ 7x = 21 \]
- Divide both sides by 7:
\[ x = \frac{21}{7} \] \[ x = 3 \]
Therefore, the value of \(x\) that proves that parallelogram ABCD is a rectangle is:
\[ \boxed{3} \]
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