To find the length of AC in the rectangle, we first need to solve for \( x \) by setting the expressions for \( AO \) and \( OD \) equal to each other, since in a rectangle, opposite sides are equal.
We have: \[ AO = OD \] \[ 9x - 1 = 6x + 17 \]
Now, let's solve for \( x \).
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Subtract \( 6x \) from both sides: \[ 9x - 6x - 1 = 17 \] \[ 3x - 1 = 17 \]
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Add \( 1 \) to both sides: \[ 3x = 18 \]
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Divide by \( 3 \): \[ x = 6 \]
Now that we have \( x \), we can find the lengths of \( AO \) and \( OD \):
Substituting \( x = 6 \) into the expression for \( AO \): \[ AO = 9(6) - 1 = 54 - 1 = 53 \]
Now substituting into the expression for \( OD \): \[ OD = 6(6) + 17 = 36 + 17 = 53 \]
Both values for \( AO \) and \( OD \) are equal, confirming our calculations.
Since \( AC \) is equal to \( AO \) (as \( AO \) and \( AC \) represent the same dimension of the rectangle), we find that: \[ AC = 53 \]
Thus, the final answer is: \[ AC = 53 \]