Question

For the rectangle shown, AO = 9x-1 and OD = 6x+17. Find AC.

Answer 1

To find the length of AC in the rectangle, we first need to understand that since AOD is a rectangle, the segments AO and OD are equal.

Given:
- AO = 9x - 1
- OD = 6x + 17

Set AO equal to OD:

\[
9x - 1 = 6x + 17
\]

Now, solve for \(x\):

1. Subtract \(6x\) from both sides:
\[
9x - 6x - 1 = 17
\]
\[
3x - 1 = 17
\]

2. Add 1 to both sides:
\[
3x = 18
\]

3. Divide by 3:
\[
x = 6
\]

Now that we have \(x\), we can find the lengths of AO and OD.

1. Calculate AO:
\[
AO = 9x - 1 = 9(6) - 1 = 54 - 1 = 53
\]

2. Calculate OD:
\[
OD = 6x + 17 = 6(6) + 17 = 36 + 17 = 53
\]

Both values confirm that AO = OD, which is expected for a rectangle.

To find AC, we can use either AO or OD since they represent the same length. Therefore:

\[
AC = AO = 53
\]

Thus, the length of AC is \(\boxed{53}\).