If A(-2, -1), B(a, 0), C(4, b) and D(1, 2) are the vertices of a parallelogram, find the values of a and b.

1. Break Down the Problem

To find the values of $a$ and $b$ such that the points A, B, C, and D form the vertices of a parallelogram, we can use the property that the diagonals of a parallelogram bisect each other. Thus, we can set up equations based on the midpoints of the diagonals AC and BD being equal.

2. Relevant Concepts

The midpoint $M$ of a segment connecting points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

3. Analysis and Detail

1. Calculate the midpoint of AC:

   • Points A and C are $A(-2, -1)$ and $C(4, b)$.
   • The midpoint $M_{AC}$ is: $M_{AC} = \left( \frac{-2 + 4}{2}, \frac{-1 + b}{2} \right) = \left( \frac{2}{2}, \frac{-1 + b}{2} \right) = (1, \frac{-1 + b}{2})$

2. Calculate the midpoint of BD:

   • Points B and D are $B(a, 0)$ and $D(1, 2)$.
   • The midpoint $M_{BD}$ is: $M_{BD} = \left( \frac{a + 1}{2}, \frac{0 + 2}{2} \right) = \left( \frac{a + 1}{2}, 1 \right)$

3. Set the midpoints equal: Now, equate the midpoints $M_{AC}$ and $M_{BD}$: $(1, \frac{-1 + b}{2}) = \left( \frac{a + 1}{2}, 1 \right)$

   This gives us two equations:

   • From the x-coordinates: $1 = \frac{a + 1}{2}$
   • From the y-coordinates: $\frac{-1 + b}{2} = 1$

4. Verify and Summarize

Solving the equations:

1. From the x-coordinates: $1 = \frac{a + 1}{2} \implies 2 = a + 1 \implies a = 1$

2. From the y-coordinates: $\frac{-1 + b}{2} = 1 \implies -1 + b = 2 \implies b = 3$

Final Answer

The values of $a$ and $b$ are: $a = 1, \quad b = 3$