Let the equations of perpendicular bisectors of sides `AC and AB of Delta ABC is x + y=3 and x - y=1` respectively Then vertex A is is (0,0)
The circumcentre of the `DeltaABC` is

Options

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

Answer

To find the circumcenter of triangle ABC with the given conditions, we will follow these steps:

### Step 1: Write down the equations of the perpendicular bisectors
The equations of the perpendicular bisectors of sides AC and AB are given as:
1. \( x + y = 3 \) (perpendicular bisector of AC)
2. \( x - y = 1 \) (perpendicular bisector of AB)

### Step 2: Solve the equations simultaneously
To find the circumcenter, we need to find the intersection of these two lines. We can solve these equations simultaneously.

From the first equation:
\[ x + y = 3 \quad \text{(1)} \]
From the second equation:
\[ x - y = 1 \quad \text{(2)} \]

### Step 3: Add the two equations
Adding equations (1) and (2):
\[ (x + y) + (x - y) = 3 + 1 \]
This simplifies to:
\[ 2x = 4 \]
Dividing both sides by 2 gives:
\[ x = 2 \]

### Step 4: Substitute x back to find y
Now, substitute \( x = 2 \) back into one of the original equations to find \( y \). We can use equation (1):
\[ 2 + y = 3 \]
Subtracting 2 from both sides gives:
\[ y = 1 \]

### Step 5: Write down the circumcenter coordinates
Thus, the coordinates of the circumcenter of triangle ABC are \( (2, 1) \).

### Final Answer:
The circumcenter of triangle ABC is \( (2, 1) \).

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