Question

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

Answer 1

To find the length of diagonal AC in rectangle ABCD, we first need to understand the relationships between the segments given in terms of the variable \( x \).

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

Since \( O \) is the intersection of the diagonals in the rectangle ABCD, point \( O \) is the midpoint of both diagonals \( AC \) and \( BD \).

In rectangles, the diagonals are equal in length, and since \( O \) is the midpoint, the lengths of \( AO \) and \( DO \) can be used to find the sides of the rectangle.

Next, observe that:
- The full length of diagonal \( AC \) can be expressed as \( AC = AO + OC \).
- Since \( O \) is the midpoint, \( AO = OC \).

Thus, we have:
\[ OC = AO = 9x - 1 \]

And because \( O \) is also the midpoint of \( BD \):
\- We can also express the diagonal segment \( DO \) in terms of \( x \):
- Since \( O \) is at the midpoint, we have:
\[ DO = OB \]

Now, we equate the lengths considering:
\[
AO + DO = AC
\]

Substituting the expressions for \( AO \) and \( DO \):
\[
(9x - 1) + (6x + 17) = AC
\]

Combine like terms:
\[
15x + 16 = AC
\]

To find \( AC \), let's solve for \( x \) using the fact that \( AO \) and \( DO \) must be equal in a rectangle setup, since both represent half of the same diagonal:
\[
AO = DO
\]
Now set the two expressions for \( AO \) and \( DO \) equal to each other:
\[
9x - 1 = 6x + 17
\]
Now, isolate \( x \):
\[
9x - 6x = 17 + 1
\]
\[
3x = 18
\]
\[
x = 6
\]

Now substitute \( x = 6 \) back into the equations to find \( AO \):
\[
AO = 9(6) - 1 = 54 - 1 = 53
\]

Now we find \( AC \):
\[
AC = 15(6) + 16 = 90 + 16 = 106
\]

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