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
X= -21/4
X=-3
X=3
X= -42/5
Answers
GPT-4o mini
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 \(\boxed{-3}\).
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 \(\boxed{-3}\).
That doesn't seem right. Are you sure the answer isn't -21/4
GPT-4o mini
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!
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!