Find the area of that triangle whose vertices are `(1,1),(-1,4)and(3,2).`

Answer

To find the area of the triangle with vertices at \( (1, 1) \), \( (-1, 4) \), and \( (3, 2) \), we can use the formula for the area of a triangle given 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:
- \( (x_1, y_1) = (1, 1) \)
- \( (x_2, y_2) = (-1, 4) \)
- \( (x_3, y_3) = (3, 2) \)

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

Substituting the values into the formula:

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

### Step 2: Simplify the expression inside the absolute value

Calculating each term:

1. \( 1(4 - 2) = 1 \times 2 = 2 \)
2. \( -1(2 - 1) = -1 \times 1 = -1 \)
3. \( 3(1 - 4) = 3 \times -3 = -9 \)

Now, substituting these values back into the area formula:

\[ \text{Area} = \frac{1}{2} \left| 2 - 1 - 9 \right| \]

### Step 3: Calculate the final expression

Calculating inside the absolute value:

\[ 2 - 1 - 9 = 2 - 1 = 1 \quad \text{and then} \quad 1 - 9 = -8 \]

Now, taking the absolute value:

\[ \left| -8 \right| = 8 \]

### Step 4: Final calculation of the area

Now, we can calculate the area:

\[ \text{Area} = \frac{1}{2} \times 8 = 4 \]

Thus, the area of the triangle is:

\[ \text{Area} = 4 \text{ square units} \]