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

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 \(A\) is \((4, 1)\).

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 \(B\) is \((0, 1)\).

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 get \(x = y - 1\).
- Substitute \(x = y - 1\) into \(x + y - 5 = 0\):
\[ (y - 1) + y - 5 = 0 \implies 2y - 6 = 0 \implies y = 3 \]
- Now substituting \(y = 3\) back to find \(x\):
\[ x = 3 - 1 = 2 \]
- So, the point \(C\) is \((2, 3)\).

### Step 2: Identify the triangle vertices
The vertices of the triangle are:
- \(A(4, 1)\)
- \(B(0, 1)\)
- \(C(2, 3)\)

### Step 3: Determine the hypotenuse
To find the circumcenter, we need to determine which side is the hypotenuse. We can check the slopes:
- The slope of line \(AB\) (between points A and B):
\[ \text{slope}_{AB} = \frac{1 - 1}{4 - 0} = 0 \quad (\text{horizontal line}) \]
- The slope of line \(BC\) (between points B and C):
\[ \text{slope}_{BC} = \frac{3 - 1}{2 - 0} = 1 \]
- The slope of line \(AC\) (between points A and C):
\[ \text{slope}_{AC} = \frac{3 - 1}{2 - 4} = -1 \]

Since \(AB\) is horizontal and \(AC\) is vertical, we can conclude that \(AC\) is perpendicular to \(AB\). Therefore, \(AC\) is the hypotenuse.

### Step 4: Find the midpoint of the hypotenuse
The circumcenter of a right triangle is the midpoint of the hypotenuse. The coordinates of the midpoint \(M\) of line segment \(AC\) can be calculated as follows:
\[ M = \left(\frac{x_A + x_C}{2}, \frac{y_A + y_C}{2}\right) = \left(\frac{4 + 2}{2}, \frac{1 + 3}{2}\right) = \left(3, 2\right) \]

### Final Answer
The coordinates of the circumcenter are \((3, 2)\).

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