The distance between the orthocentre and the circumcentre of the triangle with vertices (0, 0), (0, a) and (b, 0) is

Options

A. `sqrt(a^(2)-b^(2))//2`
B. `a+b`
C. `a-b`
D. `sqrt(a^(2)+b^(2))//2`

Answer

To find the distance between the orthocenter and the circumcenter of the triangle with vertices (0, 0), (0, a), and (b, 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, a)
- C(b, 0)

### Step 2: Determine the orthocenter
In a right triangle, the orthocenter is located at the vertex where the right angle is formed. Here, the right angle is at vertex A(0, 0). Therefore, the orthocenter (H) is:
- H(0, 0)

### Step 3: Determine the circumcenter
The circumcenter of a right triangle is the midpoint of the hypotenuse. The hypotenuse is the line segment connecting points B(0, a) and C(b, 0).

To find the midpoint (circumcenter, O) of the segment BC, we use the midpoint formula:
\[ O\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
where \( (x_1, y_1) = (0, a) \) and \( (x_2, y_2) = (b, 0) \).

Calculating the coordinates of O:
\[ O\left(\frac{0 + b}{2}, \frac{a + 0}{2}\right) = O\left(\frac{b}{2}, \frac{a}{2}\right) \]

### Step 4: Calculate the distance between the orthocenter and circumcenter
Now, we need to find the distance between points H(0, 0) and O\(\left(\frac{b}{2}, \frac{a}{2}\right)\).

Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
where \( (x_1, y_1) = (0, 0) \) and \( (x_2, y_2) = \left(\frac{b}{2}, \frac{a}{2}\right) \).

Substituting the values:
\[ d = \sqrt{\left(\frac{b}{2} - 0\right)^2 + \left(\frac{a}{2} - 0\right)^2} \]
\[ = \sqrt{\left(\frac{b}{2}\right)^2 + \left(\frac{a}{2}\right)^2} \]
\[ = \sqrt{\frac{b^2}{4} + \frac{a^2}{4}} \]
\[ = \sqrt{\frac{a^2 + b^2}{4}} \]
\[ = \frac{1}{2}\sqrt{a^2 + b^2} \]

### Final Answer
The distance between the orthocenter and the circumcenter of the triangle is:
\[ \frac{1}{2}\sqrt{a^2 + b^2} \]

---