Find the area of the triangle whose vertices are given below :
(i) (-3,-4), (-2,-7), (-1,-9)
(ii) (3,8),(-4,2), (5,1)
(iii) (2,1), (3,2), (-1, 4)

Answer

To find the area of the triangle given the vertices, we can use the formula for the area of a triangle based on its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):

\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

We will apply this formula to each part of the question.

### Part (i): Vertices (-3, -4), (-2, -7), (-1, -9)

1. **Identify the coordinates**:
- \( (x_1, y_1) = (-3, -4) \)
- \( (x_2, y_2) = (-2, -7) \)
- \( (x_3, y_3) = (-1, -9) \)

2. **Substitute into the formula**:
\[ \text{Area} = \frac{1}{2} \left| -3(-7 + 9) + (-2)(-9 + 4) + (-1)(-4 + 7) \right| \]

3. **Calculate each term**:
- First term: \(-3(-7 + 9) = -3(2) = -6\)
- Second term: \(-2(-9 + 4) = -2(-5) = 10\)
- Third term: \(-1(-4 + 7) = -1(3) = -3\)

4. **Combine the terms**:
\[ \text{Area} = \frac{1}{2} \left| -6 + 10 - 3 \right| = \frac{1}{2} \left| 1 \right| = \frac{1}{2} \]

### Part (ii): Vertices (3, 8), (-4, 2), (5, 1)

1. **Identify the coordinates**:
- \( (x_1, y_1) = (3, 8) \)
- \( (x_2, y_2) = (-4, 2) \)
- \( (x_3, y_3) = (5, 1) \)

2. **Substitute into the formula**:
\[ \text{Area} = \frac{1}{2} \left| 3(2 - 1) + (-4)(1 - 8) + 5(8 - 2) \right| \]

3. **Calculate each term**:
- First term: \(3(2 - 1) = 3(1) = 3\)
- Second term: \(-4(1 - 8) = -4(-7) = 28\)
- Third term: \(5(8 - 2) = 5(6) = 30\)

4. **Combine the terms**:
\[ \text{Area} = \frac{1}{2} \left| 3 + 28 + 30 \right| = \frac{1}{2} \left| 61 \right| = \frac{61}{2} \]

### Part (iii): Vertices (2, 1), (3, 2), (-1, 4)

1. **Identify the coordinates**:
- \( (x_1, y_1) = (2, 1) \)
- \( (x_2, y_2) = (3, 2) \)
- \( (x_3, y_3) = (-1, 4) \)

2. **Substitute into the formula**:
\[ \text{Area} = \frac{1}{2} \left| 2(2 - 4) + 3(4 - 1) + (-1)(1 - 2) \right| \]

3. **Calculate each term**:
- First term: \(2(2 - 4) = 2(-2) = -4\)
- Second term: \(3(4 - 1) = 3(3) = 9\)
- Third term: \(-1(1 - 2) = -1(-1) = 1\)

4. **Combine the terms**:
\[ \text{Area} = \frac{1}{2} \left| -4 + 9 + 1 \right| = \frac{1}{2} \left| 6 \right| = 3 \]

### Summary of Areas
- Part (i): Area = \(\frac{1}{2}\) square units
- Part (ii): Area = \(\frac{61}{2}\) square units
- Part (iii): Area = \(3\) square units