To prove that parallelogram ABCD is a rectangle, we need to find the value of \( x \) such that the diagonals \( AC \) and \( DO \) are congruent. We know:
- \( AC = 48 \)
- \( DO = 7x + 3 \)
For the parallelogram to be a rectangle, the lengths of the diagonals must be equal. This gives us the equation:
\[ AC = DO \]
Substituting the values, we get:
\[ 48 = 7x + 3 \]
Now, we can solve for \( x \):
- Subtract 3 from both sides:
\[ 48 - 3 = 7x \]
\[ 45 = 7x \]
- Divide both sides by 7:
\[ x = \frac{45}{7} \]
Since \( \frac{45}{7} \) is not an integer, it seems we're looking for integer solutions.
Now let's check the options provided a little more closely:
- If \( x = 6 \):
\[ DO = 7(6) + 3 = 42 + 3 = 45 , (not , equal , to , 48) \]
- If \( x = 4 \):
\[ DO = 7(4) + 3 = 28 + 3 = 31 , (not , equal , to , 48) \]
- If \( x = 3 \):
\[ DO = 7(3) + 3 = 21 + 3 = 24 , (not , equal , to , 48) \]
- If \( x = 8 \):
\[ DO = 7(8) + 3 = 56 + 3 = 59 , (not , equal , to , 48) \]
After going through all the options, none of the provided values for \( x \) give congruent diagonals since \( 45 \) does not match \( 48 \).
However, based on the exact equation \( 48 = 7x + 3 \), the only solution is \( x = \frac{45}{7} \).
If you need a specific value from the provided choices, none is correct based on the mathematics presented. Please check if any more info or correct conditions are missing, but the best calculation with the given figures would lead to a non-integer result.
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