Question

Question
Use the image to answer the question.

Rectangle upper A upper B upper C upper D is shown with point upper O in the center. Diagonal segments upper A upper C and upper B upper D intersect at point upper O, in the center of the rectangle. Some line segments have measurements.

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

(1 point)
Responses
which answer is correct and why
x=3
x equals 3

x=−425
x equals negative Start Fraction 42 over 5 End Fraction

x=−3
x equals negative 3

x=−214
x equals negative Start Fraction 21 over 4 End Fraction
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Answer 1

To determine the value of \( x \), we need to analyze the relationships between the segments in the rectangle \( ABCD \).

1. **Understanding the geometry**: In rectangle \( ABCD \), we know that the diagonals \( AC \) and \( BD \) intersect at point \( O \), which is the midpoint of each diagonal.

2. **Definition of segments**:
- \( CO = -3x \)
- \( BD = x + 21 \)

Since the diagonals of a rectangle bisect each other, we know that \( CO \) is half of the diagonal \( AC \) and \( BO \) is half of diagonal \( BD \).

3. **Properties of diagonals**:
- Point \( O \) is the midpoint, so \( CO = OD \) and \( AO = OB \).
- Therefore, if \( AC \) is the diagonal, we can express the entire length \( AC \) as \( 2 \times CO \) and similarly for diagonal \( BD \).

Since \( AC \) and \( BD \) are diagonals of the rectangle, they must be equal:

\[ AC = BD. \]

4. **Equating the segments**: Given the relationships we have, we know:
\[ 2CO = BD. \]

Substituting the values we have:
\[ 2(-3x) = x + 21. \]

5. **Solving for \( x \)**: This leads us to:
\[ -6x = x + 21. \]

Now, rearranging gives:
\[ -6x - x = 21 \]
\[ -7x = 21 \]
\[ x = -3. \]

Thus, the solution for \( x \) is \( -3 \).

**Final answer**: \( x = -3 \).

This corresponds to one of the answer choices provided.