Find the area of the right angled triangle whose vertices are (2,-2), (-2, 1) and (5,2). 

Options

A. `5sqrt(2)` sq. units
B. `(25)/(2)` sq. units
C. `15sqrt(2)` sq. units
D. 10 sq. units

Answer

To find the area of the right-angled triangle with vertices at (2, -2), (-2, 1), and (5, 2), we can follow these steps:

### Step 1: Identify the vertices
Let the vertices of the triangle be:
- A(2, -2)
- B(-2, 1)
- C(5, 2)

### Step 2: Calculate the lengths of the sides using the distance formula
The distance formula is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

#### Calculate AB:
Using points A(2, -2) and B(-2, 1):
\[ AB = \sqrt{((-2) - 2)^2 + (1 - (-2))^2} \]
\[ = \sqrt{(-4)^2 + (3)^2} \]
\[ = \sqrt{16 + 9} = \sqrt{25} = 5 \]

#### Calculate AC:
Using points A(2, -2) and C(5, 2):
\[ AC = \sqrt{(5 - 2)^2 + (2 - (-2))^2} \]
\[ = \sqrt{(3)^2 + (4)^2} \]
\[ = \sqrt{9 + 16} = \sqrt{25} = 5 \]

#### Calculate BC:
Using points B(-2, 1) and C(5, 2):
\[ BC = \sqrt{(5 - (-2))^2 + (2 - 1)^2} \]
\[ = \sqrt{(7)^2 + (1)^2} \]
\[ = \sqrt{49 + 1} = \sqrt{50} = 5\sqrt{2} \]

### Step 3: Identify the base and height
In a right-angled triangle, the base and height are the two sides that form the right angle. Here, we can take:
- Base = AB = 5
- Height = AC = 5

### Step 4: Calculate the area of the triangle
The area \(A\) of a right-angled triangle is given by:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
\[ A = \frac{1}{2} \times 5 \times 5 = \frac{25}{2} \]

### Final Answer
The area of the right-angled triangle is \(\frac{25}{2}\) square units.

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