Prove that the points (2,-2), (-2, 1) and (5, 2) are the vertices of a right angled triangle. Also find the area of this triangle.

Answer

To prove that the points (2, -2), (-2, 1), and (5, 2) are the vertices of a right-angled triangle, we will first calculate the lengths of the sides formed by these points and then check if they satisfy the Pythagorean theorem. After that, we will calculate the area of the triangle.

### Step 1: Calculate the lengths of the sides of the triangle

1. **Length of side AB (between points A(2, -2) and B(-2, 1))**:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
\[ = \sqrt{((-2) - 2)^2 + (1 - (-2))^2} \]
\[ = \sqrt{(-4)^2 + (3)^2} \]
\[ = \sqrt{16 + 9} = \sqrt{25} = 5 \]

2. **Length of side AC (between points A(2, -2) and C(5, 2))**:
\[ AC = \sqrt{(5 - 2)^2 + (2 - (-2))^2} \]
\[ = \sqrt{(3)^2 + (4)^2} \]
\[ = \sqrt{9 + 16} = \sqrt{25} = 5 \]

3. **Length of side BC (between points B(-2, 1) and C(5, 2))**:
\[ BC = \sqrt{(5 - (-2))^2 + (2 - 1)^2} \]
\[ = \sqrt{(7)^2 + (1)^2} \]
\[ = \sqrt{49 + 1} = \sqrt{50} \]

### Step 2: Check if the triangle is a right-angled triangle

To check if the triangle is a right-angled triangle, we will use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (longest side) is equal to the sum of the squares of the lengths of the other two sides.

Let:
- \( a = AB = 5 \)
- \( b = AC = 5 \)
- \( c = BC = \sqrt{50} \)

Now, we check if:
\[ a^2 + b^2 = c^2 \]
Calculating:
\[ 5^2 + 5^2 = (\sqrt{50})^2 \]
\[ 25 + 25 = 50 \]
\[ 50 = 50 \]

Since the equation holds true, the points (2, -2), (-2, 1), and (5, 2) form a right-angled triangle.

### Step 3: Calculate the area of the triangle

The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can take both sides \( AB \) and \( AC \) as the base and height since they are equal.

Thus, the area is:
\[ A = \frac{1}{2} \times 5 \times 5 = \frac{25}{2} \]

### Conclusion

The points (2, -2), (-2, 1), and (5, 2) are the vertices of a right-angled triangle, and the area of this triangle is \( \frac{25}{2} \) square units.

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