Question

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

a. x = -3
b. x = -42/5
c. x = -21/4
d. x = 3

Answer 1

To find the value of \( x \) given the relationships \( CO = -3x \) and \( BD = x + 21 \) in rectangle ABCD, we start by analyzing the geometric relationships.

In a rectangle, the diagonals are equal in length. Therefore, we can set the expressions for the lengths of the diagonals equal to draw a relationship:

\[
CO = \text{half the length of diagonal BD}
\]

We also know that the diagonals of rectangle ABCD are equal, and each diagonal can be represented with this formula based on the dimensions of the rectangle.

Given that \( BD \) represents the length of diagonal \( AC \) and both diagonals are equal, we have:

\[
BD = 2CO
\]

Thus, we can express this relationship as:

\[
x + 21 = 2(-3x)
\]

Now we simplify this equation:

\[
x + 21 = -6x
\]

Next, we solve for \( x \) by first moving the \( x \) terms to one side of the equation:

\[
x + 6x = -21
\]

Combine the \( x \) terms:

\[
7x = -21
\]

Now, divide both sides by \( 7 \):

\[
x = -3
\]

Thus the value of \( x \) is:

\[
\boxed{-3}
\]

This corresponds to option (a).