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.

Image Long DescriptionSegment upper C upper O is labeled negative 3 x. Segment upper B upper D is labeled x plus 21.

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

(1 point)
Responses

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

x=3
x equals 3

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

x=−3

Answer 1

To find \( x \), we start by noting that \( CO \) and \( BD \) are the same line segment within rectangle \( ABCD \), where \( O \) is the midpoint. Therefore, we can set the lengths of those segments equal to each other:

\[
CO = BD
\]

Substituting the values given in the problem:

\[
-3x = x + 21
\]

Now, we will solve for \( x \):

1. Add \( 3x \) to both sides:

\[
0 = 4x + 21
\]

2. Subtract 21 from both sides:

\[
-21 = 4x
\]

3. Divide by 4:

\[
x = -\frac{21}{4}
\]

The correct value of \( x \) is:

\[
\boxed{-\frac{21}{4}}
\]