The three vertices of a square ABCD are A(4, 3), B(-3, 3) and C(-3,-4). Find :
(i) The coordinates of D.
(ii) The area of square ABCD.

Options

A. `{:(""(i),(ii)),((-4,-4),"49 sq. units"):}`
B. `{:(""(i),(ii)),((3,-4),"25 sq. units"):}`
C. `{:(""(i),(ii)),((2,-4),"36 sq. units"):}`
D. `{:(""(i),(ii)),((4,-4),"49 sq. units"):}`

Answer

To solve the problem, we need to find the coordinates of the fourth vertex D of the square ABCD and then calculate the area of the square.

### Step 1: Identify the coordinates of points A, B, and C.
- A(4, 3)
- B(-3, 3)
- C(-3, -4)

### Step 2: Determine the coordinates of point D.
Since ABCD is a square, we can use the property that the diagonals of a square bisect each other.

1. **Find the midpoints of the diagonal AC:**
- Midpoint M of AC = \(\left(\frac{x_A + x_C}{2}, \frac{y_A + y_C}{2}\right)\)
- Substituting the coordinates:
\[ M = \left(\frac{4 + (-3)}{2}, \frac{3 + (-4)}{2}\right) = \left(\frac{1}{2}, \frac{-1}{2}\right) \]

2. **Find the coordinates of D using the midpoint of diagonal BD:**
- Let D be (x, y). The midpoint M of BD should also equal M of AC.
- Midpoint M of BD = \(\left(\frac{x_B + x_D}{2}, \frac{y_B + y_D}{2}\right)\)
- Substituting the coordinates:
\[ M = \left(\frac{-3 + x}{2}, \frac{3 + y}{2}\right) \]
- Setting the midpoints equal:
\[ \frac{-3 + x}{2} = \frac{1}{2} \quad \text{and} \quad \frac{3 + y}{2} = \frac{-1}{2} \]

3. **Solving for x:**
\[ -3 + x = 1 \implies x = 1 + 3 = 4 \]

4. **Solving for y:**
\[ 3 + y = -1 \implies y = -1 - 3 = -4 \]

Thus, the coordinates of point D are \(D(4, -4)\).

### Step 3: Calculate the area of square ABCD.
The area of a square is given by the formula:
\[ \text{Area} = \text{side}^2 \]

1. **Calculate the length of one side (AB):**
- Using the distance formula:
\[ AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} \]
- Substituting the coordinates:
\[ AB = \sqrt{(-3 - 4)^2 + (3 - 3)^2} = \sqrt{(-7)^2 + 0} = \sqrt{49} = 7 \]

2. **Calculate the area:**
\[ \text{Area} = 7^2 = 49 \]

### Final Answers:
(i) The coordinates of D are \(D(4, -4)\).
(ii) The area of square ABCD is \(49\) square units.