find the area of triangle whose vertices are (3,8),(-4,2)and (5,1)

Explanation

To find the area of the triangle with vertices at the points (3, 8), (-4, 2), and (5, 1), we can use the formula for the area of a triangle given by its vertices:

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

Where:
- \((x_1, y_1) = (3, 8)\)
- \((x_2, y_2) = (-4, 2)\)
- \((x_3, y_3) = (5, 1)\)

### Step 1: Substitute the coordinates into the formula

Substituting the values into the formula:

\[ \text{Area} = \frac{1}{2} \left| 3(2 - 1) + (-4)(1 - 8) + 5(8 - 2) \right| \]

### Step 2: Calculate each term

Calculating each term inside the absolute value:

1. \(3(2 - 1) = 3 \times 1 = 3\)
2. \((-4)(1 - 8) = -4 \times -7 = 28\)
3. \(5(8 - 2) = 5 \times 6 = 30\)

### Step 3: Combine the terms

Now, combine these results:

\[ \text{Area} = \frac{1}{2} \left| 3 + 28 + 30 \right| \]

Calculating the sum:

\[ 3 + 28 + 30 = 61 \]

### Step 4: Final calculation

Now, substitute back into the area formula:

\[ \text{Area} = \frac{1}{2} \left| 61 \right| = \frac{61}{2} \]

### Conclusion

Thus, the area of the triangle is:

\[ \text{Area} = \frac{61}{2} \text{ square units} \]

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