Key Concepts and Formulas

• Parallelogram Property: The diagonals of a parallelogram bisect each other.
• Midpoint Formula: The midpoint of a line segment joining points $(x_1, y_1)$ and $(x_2, y_2)$ is $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$.

Step-by-Step Solution

• Step 1: Define the Given Information

We are given the vertices of parallelogram $ABCD$ as $A(\alpha, \beta)$, $B(1,0)$, $C(\gamma, \delta)$, and $D(1,2)$, where $\alpha, \beta, \gamma, \delta \in \mathbb{Z}$. We need to find the value of $2(\alpha+\beta+\gamma+\delta)$.

• Step 2: Apply the Parallelogram Property

Since $ABCD$ is a parallelogram, the diagonals $AC$ and $BD$ bisect each other. This means the midpoint of $AC$ is equal to the midpoint of $BD$.

• Step 3: Calculate the Midpoint of AC

Using the midpoint formula for $A(\alpha, \beta)$ and $C(\gamma, \delta)$, the midpoint of $AC$ is: $\text{Midpoint of } AC = \left(\frac{\alpha+\gamma}{2}, \frac{\beta+\delta}{2}\right)$

• Step 4: Calculate the Midpoint of BD

Using the midpoint formula for $B(1,0)$ and $D(1,2)$, the midpoint of $BD$ is: $\text{Midpoint of } BD = \left(\frac{1+1}{2}, \frac{0+2}{2}\right) = (1,1)$

• Step 5: Equate the Midpoints

Since the midpoints of $AC$ and $BD$ are the same, we have: $\left(\frac{\alpha+\gamma}{2}, \frac{\beta+\delta}{2}\right) = (1,1)$

• Step 6: Form Equations

Equating the $x$ and $y$ coordinates, we get two equations: * Equation 1: $\frac{\alpha+\gamma}{2} = 1 \implies \alpha+\gamma = 2$ * Equation 2: $\frac{\beta+\delta}{2} = 1 \implies \beta+\delta = 2$

• Step 7: Calculate the Required Expression

We want to find $2(\alpha+\beta+\gamma+\delta)$. We can rewrite this as: $2(\alpha+\beta+\gamma+\delta) = 2((\alpha+\gamma) + (\beta+\delta))$ Substituting the values from Equation 1 and Equation 2: $2((\alpha+\gamma) + (\beta+\delta)) = 2(2 + 2) = 2(4) = 8$

Common Mistakes & Tips

• Unnecessary Information: The problem provides $AB = \sqrt{10}$ and that $A$ and $C$ lie on the line $3y = 2x+1$. These facts are not needed to solve this specific problem using the bisection of diagonals property.
• Vertex Order: The order of vertices $ABCD$ is important. Changing the order changes the diagonals.
• Integer Constraint: While not directly used here, the fact that $\alpha, \beta, \gamma, \delta$ are integers could be crucial in other variations of this problem.

Summary

By using the property that the diagonals of a parallelogram bisect each other and applying the midpoint formula, we found that $\alpha+\gamma = 2$ and $\beta+\delta = 2$. Therefore, $2(\alpha+\beta+\gamma+\delta) = 2(2+2) = 8$.

The final answer is \boxed{8}, which corresponds to option (A).