Question

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 negative 3 x=−214 x equals negative Start Fraction 21 over 4 End Fraction x=3

Answer 1

To solve for \( x \), we start by recognizing the properties of the rectangle ABCD. In a rectangle, the diagonals \( AC \) and \( BD \) are equal in length.

Given that:
- \( CO = -3x \)
- \( BD = x + 21 \)

We also know that in a rectangle, the diagonals bisect each other. Thus, \( O \) is the midpoint of both \( AC \) and \( BD \), and the lengths will be equal, that is \( AC = BD \).

However, we aren't given the length \( AC \) directly, but we can set up an equation with the information we do have regarding the diagonal \( BD \). Since we assume \( O \) divides \( BD \) into two equal segments, we set:

\[
CO = OD
\]

Since \( O \) is the midpoint of \( BD \), we can also express \( OD \) in terms of \( BD \):

\[
OD = \frac{1}{2} BD = \frac{1}{2}(x + 21)
\]

Setting these equal gives us:

\[
-3x = \frac{1}{2}(x + 21)
\]

To eliminate the fraction, we multiply both sides by \( 2 \):

\[
-6x = x + 21
\]

Now, we solve for \( x \) by moving the \( x \) term on the right side to the left side:

\[
-6x - x = 21
\]

This simplifies to:

\[
-7x = 21
\]

Next, we divide both sides by \( -7 \):

\[
x = -3
\]

Thus, the solution for \( x \) is:

\[
\boxed{-3}
\]