The coordinates of circumcentre of the triangle whose vertices are (0,0), (4,0) and (0,-6) is

Options

A. (0,0)
B. (4,0)
C. `(0,-6)`
D. `(2,-3)`

Answer

To find the coordinates of the circumcenter of the triangle with vertices at (0,0), (4,0), and (0,-6), we can follow these steps:

### Step 1: Identify the vertices of the triangle
Let the vertices of the triangle be:
- A = (0, 0)
- B = (4, 0)
- C = (0, -6)

### Step 2: Determine the hypotenuse
Since the triangle formed by these points is a right triangle (with the right angle at A), the hypotenuse will be the line segment connecting points B and C.

### Step 3: Find the midpoint of the hypotenuse
The circumcenter of a right triangle is located at the midpoint of the hypotenuse. To find the midpoint of line segment BC, we can use the midpoint formula:
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of points B and C.

### Step 4: Apply the midpoint formula
For points B (4, 0) and C (0, -6):
- \(x_1 = 4\), \(y_1 = 0\)
- \(x_2 = 0\), \(y_2 = -6\)

Now substituting these values into the midpoint formula:
\[ \text{Midpoint} = \left( \frac{4 + 0}{2}, \frac{0 + (-6)}{2} \right) \]
\[ = \left( \frac{4}{2}, \frac{-6}{2} \right) \]
\[ = (2, -3) \]

### Step 5: Conclusion
Thus, the coordinates of the circumcenter of the triangle are \((2, -3)\).

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

---