In square ABCD, A=(3,4),B=(-2,4) and C=(-2,-1). By plotting these point on a graph paper, find the co-ordinates of vertex D. Also find the area of the square.

Answer

To find the coordinates of vertex D in square ABCD and the area of the square, we can follow these steps:

### Step 1: Plot the Given Points
We are given the coordinates of points A, B, and C:
- A = (3, 4)
- B = (-2, 4)
- C = (-2, -1)

Plot these points on a graph paper.

### Step 2: Identify the Length of the Side of the Square
To find the length of the side of the square, we can calculate the distance between points A and B using the distance formula:

\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

For points A (3, 4) and B (-2, 4):
- \(x_1 = 3\), \(y_1 = 4\)
- \(x_2 = -2\), \(y_2 = 4\)

Substituting these values into the formula:

\[ AB = \sqrt{((-2) - 3)^2 + (4 - 4)^2} \]
\[ = \sqrt{(-5)^2 + 0^2} \]
\[ = \sqrt{25} \]
\[ = 5 \]

### Step 3: Find the Coordinates of Vertex D
Since ABCD is a square, the coordinates of D can be found by using the fact that the sides are equal and perpendicular.

From point C (-2, -1) to point D, we can move vertically upwards (since AB is horizontal) by the length of the side (5 units).

Starting from C (-2, -1), we can find D as follows:
- The x-coordinate of D will remain the same as C, which is -2.
- The y-coordinate will be -1 + 5 = 4.

Thus, the coordinates of D are:
- D = (-2, 4)

### Step 4: Verify the Coordinates of D
To ensure that D is correct, we can check the distance AD:
- A = (3, 4) and D = (-2, 4)

Using the distance formula:
\[ AD = \sqrt{((-2) - 3)^2 + (4 - 4)^2} \]
\[ = \sqrt{(-5)^2 + 0^2} \]
\[ = \sqrt{25} = 5 \]

### Step 5: Calculate the Area of the Square
The area of a square is given by the formula:
\[ \text{Area} = \text{side}^2 \]
Since we found the length of the side to be 5 units:
\[ \text{Area} = 5^2 = 25 \text{ square units} \]

### Final Results
- The coordinates of vertex D are (-2, 4).
- The area of the square ABCD is 25 square units.