Question

Given that rectangle ABCD as shown, find AC if AO = 9x - 1 and DO = 6x + 17.


AC = _

Answer 1

In rectangle ABCD, points A and C are opposite corners, and point O is the intersection of the diagonals (which bisect each other). Therefore, AO is half of AC and DO is half of BD. Since diagonals of a rectangle are equal and bisect each other, we can set AO equal to DO in terms of the total lengths of the diagonals:

\[
AO = DO
\]

Given:
- \( AO = 9x - 1 \)
- \( DO = 6x + 17 \)

Set these equal to each other:

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

Now, solve for \( x \):

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

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

2. This simplifies to:

\[
3x - 1 = 17
\]

3. Next, add 1 to both sides:

\[
3x = 18
\]

4. Now divide by 3:

\[
x = 6
\]

Now that we have \( x \), we can substitute it back into one of the expressions for \( AO \) to find its length:

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

Since \( AO \) is half of \( AC \), we can find \( AC \):

\[
AC = 2 \times AO = 2 \times 53 = 106
\]

Thus, the length of \( AC \) is:

\[
\boxed{106}
\]