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

Question

(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: $$\boxed{{\displaystyle 106}}$$