To find the length of diagonal AC in rectangle ABCD, we first need to understand the relationships between the segments given in terms of the variable \( x \).
Given:
- \( AO = 9x - 1 \)
- \( DO = 6x + 17 \)
Since \( O \) is the intersection of the diagonals in the rectangle ABCD, point \( O \) is the midpoint of both diagonals \( AC \) and \( BD \).
In rectangles, the diagonals are equal in length, and since \( O \) is the midpoint, the lengths of \( AO \) and \( DO \) can be used to find the sides of the rectangle.
Next, observe that:
- The full length of diagonal \( AC \) can be expressed as \( AC = AO + OC \).
- Since \( O \) is the midpoint, \( AO = OC \).
Thus, we have: \[ OC = AO = 9x - 1 \]
And because \( O \) is also the midpoint of \( BD \): - We can also express the diagonal segment \( DO \) in terms of \( x \):
- Since \( O \) is at the midpoint, we have: \[ DO = OB \]
Now, we equate the lengths considering: \[ AO + DO = AC \]
Substituting the expressions for \( AO \) and \( DO \): \[ (9x - 1) + (6x + 17) = AC \]
Combine like terms: \[ 15x + 16 = AC \]
To find \( AC \), let's solve for \( x \) using the fact that \( AO \) and \( DO \) must be equal in a rectangle setup, since both represent half of the same diagonal: \[ AO = DO \] Now set the two expressions for \( AO \) and \( DO \) equal to each other: \[ 9x - 1 = 6x + 17 \] Now, isolate \( x \): \[ 9x - 6x = 17 + 1 \] \[ 3x = 18 \] \[ x = 6 \]
Now substitute \( x = 6 \) back into the equations to find \( AO \): \[ AO = 9(6) - 1 = 54 - 1 = 53 \]
Now we find \( AC \): \[ AC = 15(6) + 16 = 90 + 16 = 106 \]
Thus, the length of diagonal \( AC \) is: \[ \boxed{106} \]
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