For a quadrilateral to be a parallelogram, opposite sides must be equal. In quadrilateral ABCD, we have the following sides:
- Side AB = \( 40x \)
- Side BC = \( 3y + 4 \)
- Side CD = \( 8x + 32 \)
- Side AD = \( 7y - 8 \)
According to the properties of a parallelogram, we can set the following equations based on opposite sides being equal:
- \( AB = CD \) \[ 40x = 8x + 32 \]
- \( BC = AD \) \[ 3y + 4 = 7y - 8 \]
Let's solve each equation.
Solving for \( x \):
Starting with the first equation: \[ 40x = 8x + 32 \] Subtract \( 8x \) from both sides: \[ 32x = 32 \] Now divide both sides by 32: \[ x = 1 \]
Solving for \( y \):
Now let's solve the second equation: \[ 3y + 4 = 7y - 8 \] Subtract \( 3y \) from both sides: \[ 4 = 4y - 8 \] Add 8 to both sides: \[ 12 = 4y \] Now divide both sides by 4: \[ y = 3 \]
Conclusion:
The values of \( x \) and \( y \) that will make quadrilateral ABCD a parallelogram are: \[ x = 1 \quad \text{and} \quad y = 3. \]
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