Question

For rectangle ABCD point O is right in the middle with A at the bottom left B at the top left C in the top right D in the bottom right.
For the rectangle shown, AO = 9x-1 and OD = 6x+17. Find AC

Answer 1

To find the length of \( AC \), we first need to find the length of \( AO \) and \( OD \) and then, since \( AC = AO + OD \), we can add them together.

Given:

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

Since point \( O \) is the midpoint of rectangle \( ABCD \), it follows that \( AO = OD \). Therefore, we can set the expressions for \( AO \) and \( OD \) equal to each other:

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

Now, we'll solve 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. Then divide both sides by \( 3 \):

\[
x = 6
\]

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

First, calculate \( AO \):

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

Next, calculate \( OD \):

\[
OD = 6x + 17 = 6(6) + 17 = 36 + 17 = 53
\]

Since both \( AO \) and \( OD \) are equal and equal to \( 53 \), we can determine the total length \( AC \):

\[
AC = AO + OD = 53 + 53 = 106
\]

Thus, the length of \( AC \) is \( \boxed{106} \).