To solve the problem, we need to find the values of \( a \) and \( b \) such that the points \((-2, 1)\), \((a, 0)\), \((4, b)\), and \((1, 2)\) form a parallelogram. We will also find the area of the parallelogram.
### Step 1: Use the property of diagonals in a parallelogram
The diagonals of a parallelogram bisect each other. Let's denote the points as follows:
- \( A(-2, 1) \)
- \( B(a, 0) \)
- \( C(4, b) \)
- \( D(1, 2) \)
The midpoints of the diagonals \( AC \) and \( BD \) should be equal.
### Step 2: Find the midpoint of diagonal \( AC \)
The midpoint \( M_{AC} \) of \( AC \) is given by:
\[
M_{AC} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{-2 + 4}{2}, \frac{1 + b}{2} \right) = \left( 1, \frac{1 + b}{2} \right)
\]
### Step 3: Find the midpoint of diagonal \( BD \)
The midpoint \( M_{BD} \) of \( BD \) is given by:
\[
M_{BD} = \left( \frac{a + 1}{2}, \frac{0 + 2}{2} \right) = \left( \frac{a + 1}{2}, 1 \right)
\]
### Step 4: Set the midpoints equal
Since the midpoints are equal, we can set the coordinates equal to each other:
1. For the x-coordinates:
\[
1 = \frac{a + 1}{2}
\]
Multiplying both sides by 2:
\[
2 = a + 1 \implies a = 1
\]
2. For the y-coordinates:
\[
\frac{1 + b}{2} = 1
\]
Multiplying both sides by 2:
\[
1 + b = 2 \implies b = 1
\]
### Step 5: Summary of values found
Thus, we have:
- \( a = 1 \)
- \( b = 1 \)
### Step 6: Find the area of the parallelogram
The area of a parallelogram can be calculated using the formula:
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
We can split the area into two triangles: \( ABD \) and \( BCD \).
#### Area of triangle \( ABD \):
Using points \( A(-2, 1) \), \( B(1, 0) \), \( D(1, 2) \):
\[
\text{Area}_{ABD} = \frac{1}{2} \left| -2(0 - 2) + 1(2 - 1) + 1(1 - 0) \right|
\]
Calculating:
\[
= \frac{1}{2} \left| -2(-2) + 1(1) + 1(1) \right| = \frac{1}{2} \left| 4 + 1 + 1 \right| = \frac{1}{2} \times 6 = 3
\]
#### Area of triangle \( BCD \):
Using points \( B(1, 0) \), \( C(4, 1) \), \( D(1, 2) \):
\[
\text{Area}_{BCD} = \frac{1}{2} \left| 1(1 - 2) + 4(2 - 0) + 1(0 - 1) \right|
\]
Calculating:
\[
= \frac{1}{2} \left| 1(-1) + 4(2) + 1(-1) \right| = \frac{1}{2} \left| -1 + 8 - 1 \right| = \frac{1}{2} \times 6 = 3
\]
### Step 7: Total area of the parallelogram
The total area of the parallelogram is:
\[
\text{Area}_{ABCD} = \text{Area}_{ABD} + \text{Area}_{BCD} = 3 + 3 = 6
\]
### Final Results
(i) \( a = 1 \)
(ii) \( b = 1 \)
(iii) Area of the parallelogram = 6 square units.
