Find the area of the triangle with vertices `(3,8),(-4,2)` and `(5,1)`.

Answer

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

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

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 coordinates into the area formula, we have:

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

### Step 2: Simplify the expression inside the absolute value
Now, we simplify the expression:

\[ = \frac{1}{2} \left| 3(1) + (-4)(-7) + 5(6) \right| \]

Calculating each term:

- \( 3(1) = 3 \)
- \( -4(-7) = 28 \)
- \( 5(6) = 30 \)

Adding these values together:

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

### Step 3: Calculate the total
Now, we calculate the total:

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

### Step 4: Final calculation
Thus, the area of the triangle is:

\[ = \frac{1}{2} \times 61 = 30.5 \]

### Final Answer
The area of the triangle is \( 30.5 \) square units.

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