The equations of the sides of a triangle are x+y-5=0, x-y+1=0, and y-1=0. Then the coordinates of the circumcenter are

Options

A. 2,1
B. 1,2
C. 2,-2
D. 1,-2

Explanation

To find the coordinates of the circumcenter of the triangle formed by the lines \(x + y - 5 = 0\), \(x - y + 1 = 0\), and \(y - 1 = 0\), we will follow these steps:

### Step 1: Find the points of intersection of the lines.
We need to find the vertices of the triangle by solving the equations of the lines pairwise.

1. **Intersection of \(x + y - 5 = 0\) and \(y - 1 = 0\)**:
- Substitute \(y = 1\) into \(x + y - 5 = 0\):
\[ x + 1 - 5 = 0 \implies x = 4 \]
- So, the point is \(A(4, 1)\).

2. **Intersection of \(x - y + 1 = 0\) and \(y - 1 = 0\)**:
- Substitute \(y = 1\) into \(x - y + 1 = 0\):
\[ x - 1 + 1 = 0 \implies x = 0 \]
- So, the point is \(B(0, 1)\).

3. **Intersection of \(x + y - 5 = 0\) and \(x - y + 1 = 0\)**:
- We can solve these two equations simultaneously. From \(x - y + 1 = 0\), we have \(y = x + 1\).
- Substitute \(y = x + 1\) into \(x + y - 5 = 0\):
\[ x + (x + 1) - 5 = 0 \implies 2x - 4 = 0 \implies x = 2 \]
- Then, substituting \(x = 2\) back to find \(y\):
\[ y = 2 + 1 = 3 \]
- So, the point is \(C(2, 3)\).

### Step 2: Identify the circumcenter.
The circumcenter of a triangle can be found as the midpoint of the hypotenuse if the triangle is a right triangle.

1. **Identify the hypotenuse**:
- The triangle formed by points \(A(4, 1)\), \(B(0, 1)\), and \(C(2, 3)\) has a right angle at point \(B\) since the lines \(AB\) and \(BC\) are perpendicular.

### Step 3: Calculate the midpoint of the hypotenuse \(AC\).
The hypotenuse is the line segment connecting points \(A(4, 1)\) and \(C(2, 3)\).

1. **Midpoint formula**:
\[ \text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
- For points \(A(4, 1)\) and \(C(2, 3)\):
\[ \text{Midpoint} = \left(\frac{4 + 2}{2}, \frac{1 + 3}{2}\right) = \left(\frac{6}{2}, \frac{4}{2}\right) = (3, 2) \]

### Conclusion:
The coordinates of the circumcenter of the triangle are \((3, 2)\).

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