Question

wxyz is a quadrilateral. wx = 2x + 16, wz = 8x + 16, xy = 2y - 16, and zy = y - 4. what values should x & y be for that wxyz is a parallelogram? use these values to determine the value

Explanation:

Step1: Recall parallelogram property

In a parallelogram, opposite sides are equal. So, \( WX = ZY \) and \( WZ = XY \). Wait, no, correction: In parallelogram \( WXYZ \), \( WX \parallel ZY \) and \( WX = ZY \), also \( WZ \parallel XY \) and \( WZ = XY \). Wait, given \( WX = 2x + 16 \), \( WZ = 8x + 16 \), \( XY = 2y - 16 \), \( ZY = y - 4 \). So for opposite sides: \( WX = ZY \)? Wait no, let's label the parallelogram: vertices \( W, X, Y, Z \) in order, so \( WX \) and \( ZY \) are opposite, \( WZ \) and \( XY \) are opposite. So \( WX = ZY \) and \( WZ = XY \). Wait, no, \( WX \) is from \( W \) to \( X \), \( ZY \) is from \( Z \) to \( Y \), so they are opposite. \( WZ \) is from \( W \) to \( Z \), \( XY \) is from \( X \) to \( Y \), opposite. So:

1. \( WX = ZY \): \( 2x + 16 = y - 4 \)
2. \( WZ = XY \): \( 8x + 16 = 2y - 16 \)

Wait, maybe I mixed up. Wait, in parallelogram, opposite sides are equal: \( WX = ZY \) and \( WZ = XY \)? Wait no, \( WX \) and \( YZ \) (ZY) are opposite, \( WZ \) and \( XY \) are opposite. So:

Equation 1: \( WX = ZY \) → \( 2x + 16 = y - 4 \)

Equation 2: \( WZ = XY \) → \( 8x + 16 = 2y - 16 \)

Now we can solve these two equations. Let's use substitution or elimination. From Equation 1: \( y = 2x + 16 + 4 = 2x + 20 \). Substitute \( y = 2x + 20 \) into Equation 2:

\( 8x + 16 = 2(2x + 20) - 16 \)

Step2: Solve for x

Expand right side: \( 8x + 16 = 4x + 40 - 16 \)

Simplify right side: \( 8x + 16 = 4x + 24 \)

Subtract \( 4x \) from both sides: \( 4x + 16 = 24 \)

Subtract 16: \( 4x = 8 \)

Divide by 4: \( x = 2 \)

Step3: Solve for y

Substitute \( x = 2 \) into \( y = 2x + 20 \): \( y = 2(2) + 20 = 4 + 20 = 24 \)

Wait, let's check with Equation 2: \( WZ = 8x + 16 = 8(2) + 16 = 16 + 16 = 32 \). \( XY = 2y - 16 = 2(24) - 16 = 48 - 16 = 32 \). Good. \( WX = 2x + 16 = 2(2) + 16 = 4 + 16 = 20 \). \( ZY = y - 4 = 24 - 4 = 20 \). Perfect, opposite sides equal.

Answer:

\( x = 2 \), \( y = 24 \)