To determine the value of \( x \) that proves parallelogram ABCD is a rectangle, we use the information given:
- The length of diagonal \( AC \) is 48.
- The length of diagonal \( DO \) is expressed as \( 7x + 3 \).
Since \( AC \) and \( DO \) are the diagonals of the parallelogram, and for a parallelogram to be a rectangle, the diagonals must be congruent:
\[ AC = DO \]
Setting the two expressions equal gives us the following equation:
\[ 48 = 7x + 3 \]
Now, we can solve for \( x \):
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Subtract 3 from both sides:
\[ 48 - 3 = 7x \]
\[ 45 = 7x \]
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Now divide both sides by 7:
\[ x = \frac{45}{7} \]
Since \( \frac{45}{7} \) does not match any of the options given (4, 8, 6, or 3), we must have misunderstood or mislabeled the problem's diagonals. However, we can conclude that we need the value of \( 7x + 3 \) to equal 48.
Let’s verify if the \( x \) values given yield a diagonal length of 48:
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If \( x = 4 \): \[ DO = 7(4) + 3 = 28 + 3 = 31 \]
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If \( x = 8 \): \[ DO = 7(8) + 3 = 56 + 3 = 59 \]
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If \( x = 6 \): \[ DO = 7(6) + 3 = 42 + 3 = 45 \]
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If \( x = 3 \): \[ DO = 7(3) + 3 = 21 + 3 = 24 \]
None of these calculated diagonal lengths are equal to 48. Therefore, based on the given solutions, it looks likely that none of the options correctly prove the condition that \( DO = 48 \).
Please verify if there were any errors in the setup or provided options, as the mathematical operations lead to no correct answers among the choices given for \( x \).
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