Step-by-step Solution:

Step 1: The given triangle has vertices: \[ O(0,0), \quad A(a,0), \quad B(0,a) \] Step 2: The circumcenter of a right-angled triangle is the midpoint of the hypotenuse. \[\] The hypotenuse is AB since \(O\) is the right angle. \[\] Midpoint of \( A(a, 0) \) and \( B(0, a) \): \[ \text{Circumcenter} \, C = \left(\frac{a+0}{2}, \frac{0+a}{2}\right) = \left(\frac{a}{2}, \frac{a}{2} \right) \] Step 3: The orthocenter of a right-angled triangle is the vertex at the right angle. \[\] Since \( O(0,0) \) is the right angle, the orthocenter is at \( O(0,0) \). Step 4: Compute the Distance between Circumcenter and Orthocenter \[\] The distance between \( (0,0) \) and \( \left(\frac{a}{2}, \frac{a}{2} \right) \) is given by: \[ d = \sqrt{\left(\frac{a}{2} - 0\right)^2 + \left(\frac{a}{2} - 0\right)^2} \] \[ = \sqrt{\left(\frac{a}{2}\right)^2 + \left(\frac{a}{2}\right)^2} \] \[ = \sqrt{\frac{a^2}{4} + \frac{a^2}{4}} \] \[ = \sqrt{\frac{2a^2}{4}} \] \[ = \sqrt{\frac{a^2}{2}} \] \[ = \frac{a}{\sqrt{2}} \]