Question

The rectangle ABCD is given as shown. Find x if CO= -3x and BD=x+21.

X= -21/4
X=-3
X=3
X= -42/5

Answer 1

That doesn't seem right. Are you sure the answer isn't -21/4

Answer 2

In rectangle ABCD, we know that the diagonals are equal in length and bisect each other. Let \(O\) be the midpoint of both diagonals \(AC\) and \(BD\).

Given the lengths:

- \(CO = -3x\)
- \(BD = x + 21\)

Since \(O\) is the midpoint of diagonal \(BD\), we can express the lengths as follows:

1. The length of diagonal \(BD\) is \(BD = 2 \cdot CO\) (because diagonal \(BD\) is twice the length of segment \(CO\) since \(O\) bisects \(BD\)).

Setting these equal gives us the equation:

\[
x + 21 = 2(-3x)
\]

Now simplify and solve for \(x\):

1. Expand the right side:
\[
x + 21 = -6x
\]

2. Add \(6x\) to both sides to combine like terms:
\[
x + 6x + 21 = 0
\]
\[
7x + 21 = 0
\]

3. Subtract 21 from both sides:
\[
7x = -21
\]

4. Divide by 7:
\[
x = -3
\]

Hence, the value of \(x\) is \(\boxed{-3}\).

Answer 3

Let's revisit the problem step by step to confirm the calculations.

We have the following equations:

1. \( CO = -3x \)
2. \( BD = x + 21 \)

Since \( O \) is the midpoint of the diagonal \( BD \), we know that \( BD = 2 \cdot CO \).

So, we can set up the equation:

\[
x + 21 = 2(-3x)
\]

Now, simplifying the right side:

\[
x + 21 = -6x
\]

Now, let's solve for \( x \):

1. Add \( 6x \) to both sides:

\[
x + 6x + 21 = 0
\]
\[
7x + 21 = 0
\]

2. Subtract 21 from both sides:

\[
7x = -21
\]

3. Divide by 7:

\[
x = -3
\]

So, the solution clearly points to \( x = -3 \).

### Conclusion
The value of \( x \) is indeed \( -3 \), confirming the earlier calculation. \( -\frac{21}{4} \) does not satisfy the given conditions in this context. If you have other constraints or interpretations of the problem, please let me know!