To determine the value of \( x \) that proves that parallelogram ABCD is a rectangle, we are given that the diagonals \( AC \) and \( DO \) are congruent.
From the information provided:
- \( AC = 48 \)
- \( DO = 7x + 3 \)
To prove that ABCD is a rectangle, we need to set the lengths of the diagonals equal to each other because in a rectangle, the diagonals are congruent:
\[ AC = DO \]
Substituting the given values:
\[ 48 = 7x + 3 \]
Now, we can solve for \( x \):
- Subtract 3 from both sides:
\[ 48 - 3 = 7x \] \[ 45 = 7x \]
- Divide both sides by 7:
\[ x = \frac{45}{7} \approx 6.43 \]
Since none of the provided response options match \( x \approx 6.43 \), it appears we should recheck the calculations or the validity of the information regarding diagonals.
However, given the responses, let's quickly evaluate the options provided to see if any values of \( x \) lead to \( DO \) equaling \( 48 \):
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For \( x = 8 \): \[ DO = 7(8) + 3 = 56 + 3 = 59 \]
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For \( x = 4 \): \[ DO = 7(4) + 3 = 28 + 3 = 31 \]
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For \( x = 6 \): \[ DO = 7(6) + 3 = 42 + 3 = 45 \]
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For \( x = 3 \): \[ DO = 7(3) + 3 = 21 + 3 = 24 \]
None of the values \( x = 8 \), \( x = 4 \), \( x = 6 \), or \( x = 3 \) yield \( DO = 48 \). Based on the previous calculations, \( \frac{45}{7} \) was derived which doesn't fit the given choices.
However, the closest value would be if we consider a concept error or alternate checks. Since the information appears to mismatch and needs re-evaluation, I suggest aiming for precise values through re-calibrated data.
Check your values again and please assess if a different value for \( AC \) could have been recording or if contextual information resolves from \( AC = DO \). Please refer back to the source if additional data was omitted in description of diagram values or redefine what your references indicate.
If you find any specific \( x \) yielding \( DO = 48 \), only then ABCD can run implications for rectangles here based on congruency conditions previously mentioned. Maintain critical checks on lengths defined as per that geometery.