The area of triangle with vertices (1, 2), (2, 7) and (4, 9) is :

Options

A. 4 sq. units
B. 8 sq. units
C. 16 sq. units
D. 32 sq. units

Answer

To find the area of the triangle with vertices at points \( A(1, 2) \), \( B(2, 7) \), and \( C(4, 9) \), 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:
- \( (x_1, y_1) = (1, 2) \)
- \( (x_2, y_2) = (2, 7) \)
- \( (x_3, y_3) = (4, 9) \)

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

Substituting the coordinates into the area formula:

\[ \text{Area} = \frac{1}{2} \left| 1(7 - 9) + 2(9 - 2) + 4(2 - 7) \right| \]

### Step 2: Calculate the expressions inside the absolute value

Calculating each term:

1. \( 1(7 - 9) = 1 \times (-2) = -2 \)
2. \( 2(9 - 2) = 2 \times 7 = 14 \)
3. \( 4(2 - 7) = 4 \times (-5) = -20 \)

Now, substituting these values back into the area formula:

\[ \text{Area} = \frac{1}{2} \left| -2 + 14 - 20 \right| \]

### Step 3: Simplify the expression

Now simplify the expression inside the absolute value:

\[ -2 + 14 - 20 = -8 \]

### Step 4: Take the absolute value and calculate the area

Taking the absolute value:

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

Now, calculate the area:

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

### Final Answer

Thus, the area of the triangle is \( 4 \) square units.

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