Find the co- ordinates of the circumcenter of the triangle whose vertices are (3, 7) , (0, 6) and (-1, 5) . Find the circumradius.

Answer

To find the coordinates of the circumcenter of the triangle with vertices at (3, 7), (0, 6), and (-1, 5), and to calculate the circumradius, we can follow these steps:

### Step 1: Define the vertices
Let the vertices of the triangle be:
- A(3, 7)
- B(0, 6)
- C(-1, 5)

### Step 2: Set up the distance equations
The circumcenter (O) of the triangle is equidistant from all three vertices. Let the coordinates of the circumcenter be (x, y). We can set up the following distance equations based on the distance formula:

1. Distance from A to O:
\[ d_1 = \sqrt{(x - 3)^2 + (y - 7)^2} \]

2. Distance from B to O:
\[ d_2 = \sqrt{(x - 0)^2 + (y - 6)^2} \]

3. Distance from C to O:
\[ d_3 = \sqrt{(x + 1)^2 + (y - 5)^2} \]

### Step 3: Equate distances
Since \(d_1 = d_2\), we can square both sides to eliminate the square root:
\[ (x - 3)^2 + (y - 7)^2 = (x - 0)^2 + (y - 6)^2 \]
Expanding both sides:
\[ (x^2 - 6x + 9 + y^2 - 14y + 49) = (x^2 + y^2 - 12y + 36) \]
Cancelling \(x^2\) and \(y^2\) from both sides:
\[ -6x + 58 = -12y + 36 \]
Rearranging gives:
\[ 6x + 2y = 22 \quad \text{(Equation 1)} \]

Now, equate \(d_2 = d_3\):
\[ (x - 0)^2 + (y - 6)^2 = (x + 1)^2 + (y - 5)^2 \]
Expanding both sides:
\[ (x^2 + y^2 - 12y + 36) = (x^2 + 2x + 1 + y^2 - 10y + 25) \]
Cancelling \(x^2\) and \(y^2\) from both sides:
\[ -12y + 36 = 2x + 1 - 10y + 25 \]
Rearranging gives:
\[ 2x + 2y = 10 \quad \text{(Equation 2)} \]

### Step 4: Solve the system of equations
Now we have two linear equations:
1. \(6x + 2y = 22\)
2. \(2x + 2y = 10\)

We can simplify Equation 2 by dividing everything by 2:
\[ x + y = 5 \quad \text{(Equation 3)} \]

Now we can substitute Equation 3 into Equation 1:
Substituting \(y = 5 - x\) into \(6x + 2(5 - x) = 22\):
\[ 6x + 10 - 2x = 22 \]
\[ 4x + 10 = 22 \]
\[ 4x = 12 \quad \Rightarrow \quad x = 3 \]

Now substituting \(x = 3\) back into Equation 3:
\[ 3 + y = 5 \quad \Rightarrow \quad y = 2 \]

Thus, the coordinates of the circumcenter are:
\[ \text{Circumcenter} = (3, 2) \]

### Step 5: Calculate the circumradius
The circumradius \(R\) is the distance from the circumcenter to any vertex. We can use vertex B(0, 6) for this calculation:
\[ R = \sqrt{(3 - 0)^2 + (2 - 6)^2} \]
Calculating:
\[ R = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Final Answer
The coordinates of the circumcenter are \((3, 2)\) and the circumradius is \(5\) units.

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