To find the length of diagonal AC in rectangle ABCD, we use the properties of rectangles and the information given.
In rectangle ABCD, points A, B, C, and D are the corners, and point O is the intersection point of the diagonals (which is also the midpoint of each diagonal). In any rectangle, the diagonals bisect each other.
Given:
- AO = \(9x - 1\)
- DO = \(6x + 17\)
Since O is the midpoint of both diagonals, AO = OC and DO = OB. In a rectangle, the lengths of these segments are equal, hence: \[ AO = DO \] So, we set the equations equal to each other: \[ 9x - 1 = 6x + 17 \]
Now, we 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 both sides by \(3\): \[ x = 6 \]
Now that we have \(x\), we substitute it back to find the lengths of AO and DO:
Calculate AO: \[ AO = 9x - 1 = 9(6) - 1 = 54 - 1 = 53 \]
Since AO = OC and DO = OB, both segments AO and DO equal 53 units.
Now we need to calculate the length of diagonal AC. In a rectangle: \[ AC = 2 \times AO = 2 \times 53 = 106 \]
Therefore, the length of diagonal AC is: \[ \boxed{106} \]
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