Question

use the image to answer the question

given that ( a b c d ) is a rectangle, ( b o = 3 x + 32 ) and ( a c = - 8 x + 8 ), which of the following is the value of ( x ) ?
(1 point)
( x = - 4 )
( x = \frac { 1 8 } { 1 9 } )
( x = \frac { 2 8 } { 1 1 } )
( x = 2 8 )
incorrect

• this is not ( x ) value when ( b o = a c )

Explanation:

Step1: Use the property of rectangle diagonals

In a rectangle \(ABCD\), the diagonals are equal and bisect each other. So \(AC = 2BO\).

Step2: Substitute the given expressions

Given \(BO=3x + 32\) and \(AC=-8x + 8\), we substitute into \(AC = 2BO\):
\(-8x + 8=2(3x + 32)\)

Step3: Expand the right - hand side

Using the distributive property \(a(b + c)=ab+ac\), we have \(2(3x + 32)=6x+64\). So the equation becomes \(-8x + 8=6x+64\)

Step4: Move the \(x\) terms to one side and constants to the other side

Add \(8x\) to both sides: \(-8x+8x + 8=6x+8x + 64\), which simplifies to \(8 = 14x+64\). Then subtract \(64\) from both sides: \(8-64=14x+64 - 64\), so \(-56 = 14x\)

Step5: Solve for \(x\)

Divide both sides by \(14\): \(x=\frac{-56}{14}=-4\)

Answer:

\(x = - 4\)