Find the area of triangle formed by 3 lines, `y=x" "x+y=2` and x Axis

Answer

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

### Step 1: Find the intersection points of the lines
1. **Identify the equations of the lines:**
- Line 1: \(y = x\)
- Line 2: \(x + y = 2\)
- Line 3: The x-axis, which is \(y = 0\)

2. **Find the intersection of \(y = x\) and \(x + y = 2\):**
- Substitute \(y = x\) into \(x + y = 2\):
\[ x + x = 2 \implies 2x = 2 \implies x = 1 \]
- Now, substitute \(x = 1\) back into \(y = x\):
\[ y = 1 \]
- So, the intersection point is \(P(1, 1)\).

3. **Find the intersection of \(x + y = 2\) and the x-axis (\(y = 0\)):**
- Substitute \(y = 0\) into \(x + y = 2\):
\[ x + 0 = 2 \implies x = 2 \]
- So, the intersection point is \(Q(2, 0)\).

4. **Find the intersection of \(y = x\) and the x-axis (\(y = 0\)):**
- Substitute \(y = 0\) into \(y = x\):
\[ 0 = x \implies x = 0 \]
- So, the intersection point is \(R(0, 0)\).

### Step 2: Identify the vertices of the triangle
The vertices of the triangle formed by the lines are:
- \(P(1, 1)\)
- \(Q(2, 0)\)
- \(R(0, 0)\)

### Step 3: Calculate the area of the triangle
The area \(A\) of a triangle formed by 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 \(P(1, 1)\), \(Q(2, 0)\), and \(R(0, 0)\):
- \(x_1 = 1, y_1 = 1\)
- \(x_2 = 2, y_2 = 0\)
- \(x_3 = 0, y_3 = 0\)

Substituting these values into the area formula:
\[ A = \frac{1}{2} \left| 1(0 - 0) + 2(0 - 1) + 0(1 - 0) \right| \]
\[ = \frac{1}{2} \left| 0 - 2 + 0 \right| = \frac{1}{2} \left| -2 \right| = \frac{1}{2} \times 2 = 1 \]

### Conclusion
The area of the triangle formed by the lines \(y = x\), \(x + y = 2\), and the x-axis is \(1\) square unit.

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