To determine the value of \( x \) that makes quadrilateral \( ABCD \) a rectangle when given the lengths of the diagonals \( AC \) and \( BD \), we need to use the property that in a rectangle, the diagonals are equal in length.
We know that:
\[ AC = x + 7 \] \[ BD = 3x - 5 \]
Setting these two expressions for the diagonals equal to each other, we have:
\[ x + 7 = 3x - 5 \]
Next, we can solve for \( x \). First, we'll subtract \( x \) from both sides:
\[ 7 = 2x - 5 \]
Then, add 5 to both sides:
\[ 12 = 2x \]
Now, divide both sides by 2:
\[ x = 6 \]
To ensure that this value is correct, we can substitute \( x = 6 \) back into the expressions for the lengths of the diagonals to check if they are equal:
Calculating \( AC \):
\[ AC = 6 + 7 = 13 \]
Calculating \( BD \):
\[ BD = 3(6) - 5 = 18 - 5 = 13 \]
Since both diagonals \( AC \) and \( BD \) are equal to 13, the diagonals confirmation is satisfied.
Thus, the value of \( x \) that makes \( ABCD \) a rectangle is
\[ \boxed{6} \]
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