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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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1 answer

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 \( BD \).

  1. 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:

\[ AC = BD. \]

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

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

  1. 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.