To determine the value of \( x \) that proves parallelogram \( ABCD \) is a rectangle, we need to set the lengths of the diagonals equal to each other since the diagonals of a rectangle are congruent.
Given:
- \( AC = 48 \)
- \( DO = 7x + 3 \)
Since \( AC \) and \( DO \) are diagonals, we can set them equal to each other:
\[ 48 = 7x + 3 \]
Now, solve for \( x \):
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Subtract 3 from both sides: \[ 48 - 3 = 7x \] \[ 45 = 7x \]
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Divide both sides by 7: \[ x = \frac{45}{7} \] \[ x \approx 6.43 \]
This means there is no integer solution among the choices given. However, it is possible there was a miscalculation or a misunderstanding while processing the question.
If \( DO \) refers to half of the diagonal split into two segments and not just one of the diagonals, please provide clarification or check any additional conditions given with the image to ascertain the correct relationship.
The value resulting from the algebra is \( 6.43 \approx 6.43 \), but if you need a whole number based on the responses provided, please confirm the relationships in the image or question context. It seems the part with \( DO \) could be misconstrued, or it could be another configuration in your provided scenario.
For the options, if required to select, the closest integer to \( 45/7 \) could be \( 6 \), yet not directly justified as a congruent solution.
Without the image and additional context, I'd suggest reviewing the relationships or settings given the mathematical derivations shared.