Question

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

Answer 1

To find the length of \( AC \) in the rectangle \( ABCD \) given \( AO = 9x - 1 \) and \( DO = 6x + 17 \), we first need to recognize some properties of rectangles.

In a rectangle, the diagonals are equal in length. Therefore, we can establish the following relationships:

1. Opposite sides of a rectangle are equal, which gives us:
\[
AB = CD \quad \text{and} \quad AD = BC
\]

2. Diagonal \( AC \) can be calculated using the lengths of the sides \( AB \) and \( AD \) via the Pythagorean theorem:
\[
AC = \sqrt{AB^2 + AD^2}
\]

Given \( O \) is the intersection of the diagonals, and since the diagonals bisect each other at point \( O \), both segments \( AO \) and \( OC \) are equal, as well as \( DO \) and \( OB \).

From the information provided:
\[
AO = OC \quad \text{and} \quad DO = OB
\]

Now, since \( AO \) and \( DO \) are segments formed by the diagonals intersecting at point \( O \), we can set \( AO = DO \):
\[
9x - 1 = 6x + 17
\]

Next, we solve for \( x \):
\[
9x - 6x = 17 + 1
\]
\[
3x = 18
\]
\[
x = 6
\]

Now that we have \( x \), we can substitute it back into the expressions for \( AO \) and \( DO \):

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

Calculating \( DO \):
\[
DO = 6(6) + 17 = 36 + 17 = 53
\]

Since both values are equal, it confirms our calculations. Thus, we know that:
\[
AO = OC = 53 \quad \text{and} \quad DO = OB = 53
\]

We can now find the length of \( AC \):
By the properties of rectangles and the lengths of the segments:
\[
AC = AO + OC = AO + AO = 53 + 53 = 106
\]

Thus, the length of diagonal \( AC \) is:
\[
\boxed{106}
\]