Find the area of the triangle, whose vertices are (2,1), (4,5) and (6,3).

Options

A. `3`
B. `6`
C. `9`
D. `12`

Explanation

To find the area of the triangle with vertices at (2, 1), (4, 5), and (6, 3), we can use the formula for the area of a triangle given its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):

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

### Step 1: Identify the coordinates
Let:
- \( (x_1, y_1) = (2, 1) \)
- \( (x_2, y_2) = (4, 5) \)
- \( (x_3, y_3) = (6, 3) \)

### Step 2: Substitute the coordinates into the formula
Substituting the values into the area formula:

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

### Step 3: Calculate each term
Now, calculate each term inside the absolute value:

1. \( 2(5 - 3) = 2 \times 2 = 4 \)
2. \( 4(3 - 1) = 4 \times 2 = 8 \)
3. \( 6(1 - 5) = 6 \times (-4) = -24 \)

### Step 4: Combine the results
Now combine these results:

\[ \text{Area} = \frac{1}{2} \left| 4 + 8 - 24 \right| = \frac{1}{2} \left| -12 \right| \]

### Step 5: Calculate the area
Taking the absolute value and dividing by 2:

\[ \text{Area} = \frac{1}{2} \times 12 = 6 \]

### Final Answer
Thus, the area of the triangle is \( 6 \) square units.

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