Find the area of a rhombus ABCD whose vertices are A(3,0) ,B(4,5),C(-1,4) and D(-2,-1)

Answer

To find the area of the rhombus ABCD with given vertices A(3,0), B(4,5), C(-1,4), and D(-2,-1), we will follow these steps:

### Step 1: Identify the diagonals
The diagonals of the rhombus are AC and BD.

### Step 2: Calculate the length of diagonal AC
Using the distance formula:
\[ AC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
where \( A(3,0) \) and \( C(-1,4) \).

Substituting the coordinates:
\[ AC = \sqrt{((-1) - 3)^2 + (4 - 0)^2} \]
\[ = \sqrt{(-4)^2 + (4)^2} \]
\[ = \sqrt{16 + 16} \]
\[ = \sqrt{32} \]
\[ = 4\sqrt{2} \]

### Step 3: Calculate the length of diagonal BD
Using the distance formula again for points B(4,5) and D(-2,-1):
\[ BD = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ BD = \sqrt{((-2) - 4)^2 + ((-1) - 5)^2} \]
\[ = \sqrt{(-6)^2 + (-6)^2} \]
\[ = \sqrt{36 + 36} \]
\[ = \sqrt{72} \]
\[ = 6\sqrt{2} \]

### Step 4: Calculate the area of the rhombus
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times AC \times BD \]
Substituting the values of AC and BD:
\[ A = \frac{1}{2} \times (4\sqrt{2}) \times (6\sqrt{2}) \]
\[ = \frac{1}{2} \times 24 \times 2 \]
\[ = \frac{48}{2} \]
\[ = 24 \]

### Final Answer
The area of the rhombus ABCD is \( 24 \) square units.

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