Question

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

Answer 1

In a rectangle, opposite sides are equal in length. Therefore, we have the following equations based on the provided information:

\[
AO = OD
\]

Substituting the expressions given for AO and OD:

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

Now, we will solve this equation for \( x \).

1. Subtract \( 6x \) from both sides:

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

This simplifies to:

\[
3x - 1 = 17
\]

2. Next, add 1 to both sides:

\[
3x = 18
\]

3. Finally, divide by 3 to find \( x \):

\[
x = 6
\]

Now that we have \( x \), we can substitute it back into either expression to find \( AO \) or \( OD \). We'll calculate \( AO \):

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

Since \( AO = OD \), we know:

\[
OD = 53
\]

Now, to find \( AC \), we recognize that in a rectangle, the lengths of adjacent sides are different. Thus, if \( AO \) corresponds to one length of the rectangle, \( AC \) would correspond to the width, which is the same as \( DO \).

Since both \( AO \) and \( OD \) represent the lengths of the sides, we can conclude one of the sides \( AC = OD = 53 \).

Thus, the length \( AC \) is:

\[
\boxed{53}
\]