If the vertices of a triangle are at `O(0, 0), A (a, 0) and B (0, a)`. Then, the distance between its circumcentre and orthocentre is

Options

A. `(a)/(2)`
B. `(a)/(sqrt(2))`
C. `sqrt(2)a`
D. `(a)/(4)`

Answer

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

### Step 1: Identify the Triangle Vertices
The vertices of the triangle are:
- \( O(0, 0) \) (the origin)
- \( A(a, 0) \) (on the x-axis)
- \( B(0, a) \) (on the y-axis)

### Step 2: Determine the Type of Triangle
Since the triangle is formed by the x-axis and y-axis, it is a right-angled triangle with the right angle at \( O \).

### Step 3: Find the Orthocenter
In a right-angled triangle, the orthocenter is located at the vertex where the right angle is formed. Therefore, the orthocenter \( H \) is at point \( O(0, 0) \).

### Step 4: Find the Circumcenter
The circumcenter of a right-angled triangle is located at the midpoint of the hypotenuse. The hypotenuse is the line segment \( AB \).

#### Step 4.1: Calculate the Midpoint of \( AB \)
The coordinates of points \( A \) and \( B \) are:
- \( A(a, 0) \)
- \( B(0, a) \)

The midpoint \( M \) of \( AB \) can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{a + 0}{2}, \frac{0 + a}{2} \right) = \left( \frac{a}{2}, \frac{a}{2} \right) \]

Thus, the circumcenter \( C \) is at \( \left( \frac{a}{2}, \frac{a}{2} \right) \).

### Step 5: Calculate the Distance between the Circumcenter and Orthocenter
To find the distance \( d \) between the circumcenter \( C \left( \frac{a}{2}, \frac{a}{2} \right) \) and the orthocenter \( H(0, 0) \), we use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of \( C \) and \( H \):
\[ d = \sqrt{\left( \frac{a}{2} - 0 \right)^2 + \left( \frac{a}{2} - 0 \right)^2} \]
\[ d = \sqrt{\left( \frac{a}{2} \right)^2 + \left( \frac{a}{2} \right)^2} \]
\[ d = \sqrt{\frac{a^2}{4} + \frac{a^2}{4}} = \sqrt{\frac{2a^2}{4}} = \sqrt{\frac{a^2}{2}} = \frac{a}{\sqrt{2}} \]

### Final Answer
The distance between the circumcenter and orthocenter is:
\[ \frac{a}{\sqrt{2}} \]