To determine the value of \( x \) that proves that parallelogram ABCD is a rectangle by showing that the diagonals are congruent, we need to set up the equation based on the information given.
Given:
- \( AC = 48 \)
- \( DO = 7x + 3 \)
In a parallelogram, the diagonals bisect each other, so \( AC \) and \( DO \) must be equal when the parallelogram is a rectangle.
Since \( O \) is the midpoint of both diagonals, we have: \[ AC = 2 \times AO \] Since \( DO \) is also divided into two equal segments (DO = 2 * OD), we have: \[ DO = 2 \times OD \]
We can set \( DO \) equal to \( AC \) because they are equal when the parallelogram is a rectangle: \[ 48 = 7x + 3 \]
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: \[ x = \frac{45}{7} \approx 6.43 \]
However, upon reconsidering the values provided as options, you will want to check against:
- \( 7x + 3 = 48 \)
This simplifies our earlier attempts, so using the options, we can test to see which value satisfies it:
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For \( x = 6 \): \[ 7(6) + 3 = 42 + 3 = 45 \quad (\text{Incorrect}) \]
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For \( x = 4 \): \[ 7(4) + 3 = 28 + 3 = 31 \quad (\text{Incorrect}) \]
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For \( x = 8 \): \[ 7(8) + 3 = 56 + 3 = 59 \quad (\text{Incorrect}) \]
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Lastly, for \( x = 3 \): \[ 7(3) + 3 = 21 + 3 = 24 \quad (\text{Incorrect}) \]
It appears there's a miscalculation. Therefore, if \( 7x = 48 - 3 \implies x = 6.43 \).
No provided options work as \( 6.43 \) doesn't match the solutions from provided options.
Thus, if we look closely through \( AC = 2AO \text{ and } DO = 2OD \), the answers provided likely intend to convey certain values without necessarily checking against the equality moved. Given your provided values, none suit as final confirmations to a parallelogram defined as a rectangle on those values.
Conclusion: None of the given options are correct based on the values provided that \( DO = 7x + 3\) compared to the diagonals' relations.