The circumcentre of the triangle formed by lines `x=0,y=0` and `x+y-1=0` is

Options

A. (0,0)
B. `(1/2,1/2)`
C. (1,1)
D. `(1/3,1/3)`

Answer

To find the circumcenter of the triangle formed by the lines \( x = 0 \), \( y = 0 \), and \( x + y - 1 = 0 \), we can follow these steps:

### Step 1: Identify the vertices of the triangle
The lines intersect to form a triangle. We need to find the points of intersection of these lines:

1. **Intersection of \( x = 0 \) and \( y = 0 \)**:
- This gives the point \( (0, 0) \).

2. **Intersection of \( x = 0 \) and \( x + y - 1 = 0 \)**:
- Substituting \( x = 0 \) into \( x + y - 1 = 0 \) gives \( y = 1 \).
- This gives the point \( (0, 1) \).

3. **Intersection of \( y = 0 \) and \( x + y - 1 = 0 \)**:
- Substituting \( y = 0 \) into \( x + y - 1 = 0 \) gives \( x = 1 \).
- This gives the point \( (1, 0) \).

Thus, the vertices of the triangle are \( A(0, 0) \), \( B(0, 1) \), and \( C(1, 0) \).

### Step 2: Find the circumcenter
The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. However, for a right-angled triangle (which this is, since it has a right angle at \( A(0, 0) \)), the circumcenter is located at the midpoint of the hypotenuse.

1. **Identify the hypotenuse**:
- The hypotenuse is the line segment connecting points \( B(0, 1) \) and \( C(1, 0) \).

2. **Find the midpoint of \( BC \)**:
- The midpoint \( M \) of segment \( BC \) can be found using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
- Here, \( (x_1, y_1) = (0, 1) \) and \( (x_2, y_2) = (1, 0) \).
- Thus,
\[ M = \left( \frac{0 + 1}{2}, \frac{1 + 0}{2} \right) = \left( \frac{1}{2}, \frac{1}{2} \right) \]

### Conclusion
The circumcenter of the triangle formed by the lines \( x = 0 \), \( y = 0 \), and \( x + y - 1 = 0 \) is \( \left( \frac{1}{2}, \frac{1}{2} \right) \).