Area of triangle formed by straight lines `x +y = 4, 2x - y =2 and x -2 = 0` is

Options

A. 4 sq. unit
B. 9 sq. unit
C. `7/2` sq. unit
D. None of these

Answer

To find the area of the triangle formed by the lines \(x + y = 4\), \(2x - y = 2\), and \(x - 2 = 0\), we will follow these steps:

### Step 1: Find the intersection points of the lines

1. **Line 1:** \(x + y = 4\)
- If \(x = 0\), then \(y = 4\) (Point A: (0, 4))
- If \(y = 0\), then \(x = 4\) (Point B: (4, 0))

2. **Line 2:** \(2x - y = 2\)
- If \(x = 0\), then \(y = -2\) (Point C: (0, -2))
- If \(y = 0\), then \(2x = 2 \Rightarrow x = 1\) (Point D: (1, 0))

3. **Line 3:** \(x - 2 = 0\)
- This line is vertical at \(x = 2\).

### Step 2: Find the intersection of the first two lines

To find the intersection of \(x + y = 4\) and \(2x - y = 2\), we can solve these equations simultaneously.

1. From \(x + y = 4\), we can express \(y\) in terms of \(x\):
\[ y = 4 - x \]

2. Substitute \(y\) in the second equation:
\[ 2x - (4 - x) = 2 \]
\[ 2x - 4 + x = 2 \]
\[ 3x - 4 = 2 \Rightarrow 3x = 6 \Rightarrow x = 2 \]

3. Substitute \(x = 2\) back into \(y = 4 - x\):
\[ y = 4 - 2 = 2 \]
So, the intersection point is (2, 2).

### Step 3: Identify the vertices of the triangle

The vertices of the triangle formed by the lines are:
- Point A: (0, 4)
- Point B: (4, 0)
- Point C: (2, 2)

### Step 4: Calculate the area of the triangle

The area \(A\) of a triangle given 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 points:
- \((x_1, y_1) = (0, 4)\)
- \((x_2, y_2) = (4, 0)\)
- \((x_3, y_3) = (2, 2)\)

We get:
\[ A = \frac{1}{2} \left| 0(0 - 2) + 4(2 - 4) + 2(4 - 0) \right| \]
\[ A = \frac{1}{2} \left| 0 + 4(-2) + 2(4) \right| \]
\[ A = \frac{1}{2} \left| 0 - 8 + 8 \right| = \frac{1}{2} \left| 0 \right| = 0 \]

### Conclusion

Since the area calculated is 0, this indicates that the lines do not form a triangle. Therefore, the answer is **none of these**.

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