Question

image long description
the rectangle ( abcd ) is given as shown. find ( x ) if ( co = -3x ) and ( bd = x + 21 ).
(1 point)
( \bigcirc x = -\frac{21}{4} )
( \bigcirc x = -3 )
( \bigcirc x = -\frac{42}{5} )
( \bigcirc x = 3 )

Explanation:

Step1: Recall property of rectangle diagonals

In a rectangle, the diagonals are equal and bisect each other. So, the length of the diagonal \( BD \) is twice the length of \( CO \) (since \( O \) is the midpoint of the diagonals). So we have the equation \( BD = 2\times CO \).

Step2: Substitute the given expressions

Given \( CO=-3x \) and \( BD = x + 21 \), substitute into the equation from Step 1: \( x + 21=2\times(-3x) \).

Step3: Solve the linear equation

Simplify the right - hand side: \( x + 21=-6x \).
Add \( 6x \) to both sides: \( x+6x + 21=-6x + 6x \), which gives \( 7x+21 = 0 \).
Subtract 21 from both sides: \( 7x+21-21=0 - 21 \), so \( 7x=-21 \).
Divide both sides by 7: \( x=\frac{-21}{7}=-3 \).

Answer:

\( x = - 3 \) (corresponding to the option \( x=-3 \))