Find the area (in sq units) of the triangle lines ` x - 3 y = 0, x - y = 4 and x + y = 4`

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the area of the triangle formed by the lines \( x - 3y = 0 \), \( x - y = 4 \), and \( x + y = 4 \), we will follow these steps:

### Step 1: Find the intersection points of the lines
We need to find the points where the lines intersect, as these will be the vertices of the triangle.

1. **Intersection of \( x - 3y = 0 \) and \( x - y = 4 \)**:
- From \( x - 3y = 0 \), we can express \( x \) as \( x = 3y \).
- Substitute \( x = 3y \) into \( x - y = 4 \):
\[ 3y - y = 4 \implies 2y = 4 \implies y = 2 \]
- Now substitute \( y = 2 \) back into \( x = 3y \):
\[ x = 3 \times 2 = 6 \]
- So, the intersection point is \( (6, 2) \).

2. **Intersection of \( x - 3y = 0 \) and \( x + y = 4 \)**:
- Again, substitute \( x = 3y \) into \( x + y = 4 \):
\[ 3y + y = 4 \implies 4y = 4 \implies y = 1 \]
- Substitute \( y = 1 \) back into \( x = 3y \):
\[ x = 3 \times 1 = 3 \]
- So, the intersection point is \( (3, 1) \).

3. **Intersection of \( x - y = 4 \) and \( x + y = 4 \)**:
- Solve the system:
\[ x - y = 4 \quad (1) \]
\[ x + y = 4 \quad (2) \]
- Adding (1) and (2):
\[ 2x = 8 \implies x = 4 \]
- Substitute \( x = 4 \) back into (2):
\[ 4 + y = 4 \implies y = 0 \]
- So, the intersection point is \( (4, 0) \).

### Step 2: Identify the vertices of the triangle
The vertices of the triangle are:
- A: \( (6, 2) \)
- B: \( (3, 1) \)
- C: \( (4, 0) \)

### Step 3: Use the formula for the area of a triangle
The area \( A \) of a triangle given its vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) can be calculated using the formula:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

Substituting the coordinates of the vertices:
- \( (x_1, y_1) = (6, 2) \)
- \( (x_2, y_2) = (3, 1) \)
- \( (x_3, y_3) = (4, 0) \)

We have:
\[ A = \frac{1}{2} \left| 6(1 - 0) + 3(0 - 2) + 4(2 - 1) \right| \]
Calculating each term:
\[ = \frac{1}{2} \left| 6 \cdot 1 + 3 \cdot (-2) + 4 \cdot 1 \right| \]
\[ = \frac{1}{2} \left| 6 - 6 + 4 \right| \]
\[ = \frac{1}{2} \left| 4 \right| = \frac{4}{2} = 2 \]

### Final Answer
The area of the triangle is \( 2 \) square units.

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