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 use the properties of rectangles and the information given.

In rectangle ABCD, points A, B, C, and D are the corners, and point O is the intersection point of the diagonals (which is also the midpoint of each diagonal). In any rectangle, the diagonals bisect each other.

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

Since O is the midpoint of both diagonals, AO = OC and DO = OB. In a rectangle, the lengths of these segments are equal, hence:
\[
AO = DO
\]
So, we set the equations equal to each other:
\[
9x - 1 = 6x + 17
\]

Now, we solve for \(x\):
1. Subtract \(6x\) from both sides:
\[
9x - 6x - 1 = 17
\]
\[
3x - 1 = 17
\]

2. Add \(1\) to both sides:
\[
3x = 18
\]

3. Divide both sides by \(3\):
\[
x = 6
\]

Now that we have \(x\), we substitute it back to find the lengths of AO and DO:

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

Since AO = OC and DO = OB, both segments AO and DO equal 53 units.

Now we need to calculate the length of diagonal AC. In a rectangle:
\[
AC = 2 \times AO = 2 \times 53 = 106
\]

Therefore, the length of diagonal AC is:
\[
\boxed{106}
\]