The sides of triangle ABC are represented by `y-x =2, 2x +y + 4=0 and x + 2y =4.` The coordinates of circumentre are

Options

A. `(- (7)/(3), (7)/(3))`
B. `((1)/(3), (2)/(3))`
C. `(-(1)/(3), (2)/(3))`
D. none of these

Answer

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

### Step 1: Find the intersection points of the lines to determine the vertices of the triangle.

1. **Find point A (intersection of \(y - x = 2\) and \(x + 2y = 4\))**:
- Rewrite the first equation: \(y = x + 2\).
- Substitute \(y\) in the second equation:
\[ x + 2(x + 2) = 4 \implies x + 2x + 4 = 4 \implies 3x + 4 = 4 \implies 3x = 0 \implies x = 0 \]
- Substitute \(x = 0\) back into \(y = x + 2\):
\[ y = 0 + 2 = 2 \]
- Thus, point A is \((0, 2)\).

2. **Find point B (intersection of \(y - x = 2\) and \(2x + y + 4 = 0\))**:
- Rewrite the first equation: \(y = x + 2\).
- Substitute \(y\) in the second equation:
\[ 2x + (x + 2) + 4 = 0 \implies 2x + x + 2 + 4 = 0 \implies 3x + 6 = 0 \implies 3x = -6 \implies x = -2 \]
- Substitute \(x = -2\) back into \(y = x + 2\):
\[ y = -2 + 2 = 0 \]
- Thus, point B is \((-2, 0)\).

3. **Find point C (intersection of \(2x + y + 4 = 0\) and \(x + 2y = 4\))**:
- Rewrite the second equation: \(y = 4 - x/2\).
- Substitute \(y\) in the first equation:
\[ 2x + (4 - 2y) + 4 = 0 \implies 2x + 4 - 2y + 4 = 0 \implies 2x - 2y + 8 = 0 \implies 2x - 2(4 - x/2) + 8 = 0 \]
- Solving gives:
\[ 2x - 8 + x = 0 \implies 3x = 8 \implies x = -4 \]
- Substitute \(x = -4\) back into \(x + 2y = 4\):
\[ -4 + 2y = 4 \implies 2y = 8 \implies y = 4 \]
- Thus, point C is \((-4, 4)\).

### Step 2: Calculate the circumcenter of triangle ABC.

The circumcenter \(O\) can be calculated using the formula:
\[ O_x = \frac{x_A + x_B + x_C}{3}, \quad O_y = \frac{y_A + y_B + y_C}{3} \]
Substituting the coordinates of points A, B, and C:
- \(A(0, 2)\)
- \(B(-2, 0)\)
- \(C(-4, 4)\)

Calculating \(O_x\):
\[ O_x = \frac{0 + (-2) + (-4)}{3} = \frac{-6}{3} = -2 \]

Calculating \(O_y\):
\[ O_y = \frac{2 + 0 + 4}{3} = \frac{6}{3} = 2 \]

Thus, the coordinates of the circumcenter are \((-2, 2)\).

### Final Answer:
The coordinates of the circumcenter are \((-2, 2)\).

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