Question

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

(1 point)
AC=

Answer 1

To find the length of diagonal \( AC \) in rectangle \( ABCD \), we can use the following properties:

1. In a rectangle, the diagonals are equal in length.
2. The diagonals bisect each other.

From your problem, \( O \) is the midpoint of the diagonals \( AC \) and \( BD \).

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

Since \( AO \) and \( DO \) are segments of the diagonal and \( O \) is the midpoint, we have \( AO = OD \).

Setting the two expressions equal to each other, we get:

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

Now, we can 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. Finally, divide both sides by 3:
\[
x = 6
\]

Now that we have the value of \( x \), we can find \( AO \) using either expression:

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

Since \( O \) is the midpoint, \( AC \) is twice \( AO \):
\[
AC = 2 \times AO = 2 \times 53 = 106
\]

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

\[
\boxed{106}
\]