The vertices of a triangle are A(0, 0), B(0, 2) and C(2, 0). The distance between circumcentre and orthocentre is

Options

A. `sqrt(2)`
B. `(1)/(sqrt(2))`
C. 2
D. `(1)/(2)`

Answer

To find the distance between the circumcenter and orthocenter of the triangle with vertices A(0, 0), B(0, 2), and C(2, 0), we can follow these steps:

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

### Step 2: Determine the type of triangle
Since the triangle has one angle that is 90 degrees (angle A), it is a right triangle. In a right triangle, the circumcenter is located at the midpoint of the hypotenuse.

### Step 3: Find the hypotenuse
The hypotenuse of triangle ABC is the line segment BC. We can find the coordinates of points B and C:
- B(0, 2)
- C(2, 0)

### Step 4: Calculate the midpoint of the hypotenuse
The midpoint M of line segment BC can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of B and C:
\[ M = \left( \frac{0 + 2}{2}, \frac{2 + 0}{2} \right) = \left( 1, 1 \right) \]

### Step 5: Determine the orthocenter
In a right triangle, the orthocenter is located at the vertex where the right angle is formed. Therefore, the orthocenter H is at point A(0, 0).

### Step 6: Calculate the distance between the circumcenter and orthocenter
Now we need to find the distance between the circumcenter M(1, 1) and the orthocenter H(0, 0). We can use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of M and H:
\[ d = \sqrt{(1 - 0)^2 + (1 - 0)^2} = \sqrt{1^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Conclusion
The distance between the circumcenter and the orthocenter is \( \sqrt{2} \).

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