Find the circum-centre of the triangle whose vertices are `A (-3 , -1) , B( 1,2)` and `C( 0 , -4)`.

Answer

To find the circumcenter of the triangle with vertices \( A(-3, -1) \), \( B(1, 2) \), and \( C(0, -4) \), we can follow these steps:

### Step 1: Find the midpoints of two sides of the triangle.
We can find the midpoints of sides \( AB \) and \( AC \).

- **Midpoint of \( AB \)**:
\[ M_{AB} = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right) = \left( \frac{-3 + 1}{2}, \frac{-1 + 2}{2} \right) = \left( \frac{-2}{2}, \frac{1}{2} \right) = (-1, 0.5) \]

- **Midpoint of \( AC \)**:
\[ M_{AC} = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) = \left( \frac{-3 + 0}{2}, \frac{-1 + (-4)}{2} \right) = \left( \frac{-3}{2}, \frac{-5}{2} \right) = \left(-1.5, -2.5\right) \]

### Step 2: Find the slopes of the sides \( AB \) and \( AC \).
- **Slope of \( AB \)**:
\[ m_{AB} = \frac{y_B - y_A}{x_B - x_A} = \frac{2 - (-1)}{1 - (-3)} = \frac{3}{4} \]

- **Slope of \( AC \)**:
\[ m_{AC} = \frac{y_C - y_A}{x_C - x_A} = \frac{-4 - (-1)}{0 - (-3)} = \frac{-3}{3} = -1 \]

### Step 3: Find the slopes of the perpendicular bisectors.
- **Slope of the perpendicular bisector of \( AB \)**:
\[ m_{perpendicular \, AB} = -\frac{1}{m_{AB}} = -\frac{4}{3} \]

- **Slope of the perpendicular bisector of \( AC \)**:
\[ m_{perpendicular \, AC} = -\frac{1}{m_{AC}} = 1 \]

### Step 4: Write the equations of the perpendicular bisectors.
Using the point-slope form of the equation of a line \( y - y_1 = m(x - x_1) \):

- **Equation of the perpendicular bisector of \( AB \)** (using midpoint \( M_{AB}(-1, 0.5) \)):
\[ y - 0.5 = -\frac{4}{3}(x + 1) \]
Simplifying:
\[ y - 0.5 = -\frac{4}{3}x - \frac{4}{3} \]
\[ y = -\frac{4}{3}x - \frac{4}{3} + \frac{3}{6} \]
\[ y = -\frac{4}{3}x - \frac{8}{6} + \frac{3}{6} \]
\[ y = -\frac{4}{3}x - \frac{5}{6} \]

- **Equation of the perpendicular bisector of \( AC \)** (using midpoint \( M_{AC}(-1.5, -2.5) \)):
\[ y + 2.5 = 1(x + 1.5) \]
Simplifying:
\[ y + 2.5 = x + 1.5 \]
\[ y = x + 1.5 - 2.5 \]
\[ y = x - 1 \]

### Step 5: Solve the equations of the perpendicular bisectors.
Set the two equations equal to each other:
\[ -\frac{4}{3}x - \frac{5}{6} = x - 1 \]
Multiply through by 6 to eliminate the fractions:
\[ -8x - 5 = 6x - 6 \]
Combine like terms:
\[ -8x - 6x = -6 + 5 \]
\[ -14x = -1 \]
\[ x = \frac{1}{14} \]

### Step 6: Substitute \( x \) back to find \( y \).
Using \( y = x - 1 \):
\[ y = \frac{1}{14} - 1 = \frac{1}{14} - \frac{14}{14} = -\frac{13}{14} \]

### Final Answer:
The circumcenter of the triangle is:
\[ \left( \frac{1}{14}, -\frac{13}{14} \right) \]

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