The distance between the circumcenter and the orthocentre of the triangle whose vertices are `(0,0),(6,8),` and `(-4,3)` is `Ldot` Then the value of `2/(sqrt(5))L` is_________

Explanation

To solve the problem, we need to find the distance between the circumcenter and the orthocenter of the triangle with vertices at \( A(0,0) \), \( B(6,8) \), and \( C(-4,3) \).

### Step-by-step Solution:

1. **Calculate the lengths of the sides of the triangle:**
- Length \( AB \):
\[ AB = \sqrt{(6 - 0)^2 + (8 - 0)^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]
- Length \( BC \):
\[ BC = \sqrt{(6 - (-4))^2 + (8 - 3)^2} = \sqrt{(6 + 4)^2 + (8 - 3)^2} = \sqrt{10^2 + 5^2} = \sqrt{100 + 25} = \sqrt{125} = 5\sqrt{5} \]
- Length \( AC \):
\[ AC = \sqrt{(0 - (-4))^2 + (0 - 3)^2} = \sqrt{(0 + 4)^2 + (0 - 3)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

2. **Determine if the triangle is a right triangle:**
- Check if \( BC^2 = AB^2 + AC^2 \):
\[ (5\sqrt{5})^2 = (10)^2 + (5)^2 \implies 125 = 100 + 25 \implies 125 = 125 \]
- Therefore, triangle \( ABC \) is a right triangle with \( C \) as the right angle.

3. **Find the circumcenter:**
- The circumcenter of a right triangle is the midpoint of the hypotenuse \( BC \):
\[ \text{Midpoint of } BC = \left( \frac{6 + (-4)}{2}, \frac{8 + 3}{2} \right) = \left( \frac{2}{2}, \frac{11}{2} \right) = (1, \frac{11}{2}) \]

4. **Find the orthocenter:**
- The orthocenter of a right triangle is the vertex at the right angle, which is point \( A(0,0) \).

5. **Calculate the distance between the circumcenter and the orthocenter:**
- Distance \( L \) between points \( (1, \frac{11}{2}) \) and \( (0, 0) \):
\[ L = \sqrt{(1 - 0)^2 + \left(\frac{11}{2} - 0\right)^2} = \sqrt{1^2 + \left(\frac{11}{2}\right)^2} = \sqrt{1 + \frac{121}{4}} = \sqrt{\frac{4 + 121}{4}} = \sqrt{\frac{125}{4}} = \frac{\sqrt{125}}{2} = \frac{5\sqrt{5}}{2} \]

6. **Calculate the value of \( \frac{2}{\sqrt{5}} L \):**
- Substitute \( L \):
\[ \frac{2}{\sqrt{5}} L = \frac{2}{\sqrt{5}} \cdot \frac{5\sqrt{5}}{2} = 5 \]

### Final Answer:
The value of \( \frac{2}{\sqrt{5}} L \) is \( 5 \).

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