What is the distance between circumcentre and orthocentre of a right angled triangle ?

Options

A. Equal to hypotenuse
B. Half to hypotenuse
C. One third to hypotenuse
D. Two third to hypotenuse

Answer

To find the distance between the circumcenter and the orthocenter of a right-angled triangle, we can follow these steps:

### Step 1: Identify the properties of a right-angled triangle
In a right-angled triangle, the circumcenter (the center of the circumcircle) is located at the midpoint of the hypotenuse. The orthocenter (the point where the altitudes intersect) is located at the vertex of the right angle.

### Step 2: Define the triangle
Let's denote the right-angled triangle as ABC, where angle C is the right angle. The hypotenuse is AB.

### Step 3: Locate the circumcenter
Since the circumcenter is at the midpoint of the hypotenuse AB, we can denote this point as O. If the length of the hypotenuse AB is denoted as 'c', then the coordinates of O can be found as follows:
- If A is at (0, 0) and B is at (c, 0), then O will be at (c/2, 0).

### Step 4: Locate the orthocenter
The orthocenter H of triangle ABC is at point C, which is the vertex of the right angle. If C is at (0, h) where h is the height from C to the hypotenuse, then H is at (0, h).

### Step 5: Calculate the distance between O and H
To find the distance between the circumcenter O and the orthocenter H, we can use the distance formula:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of O and H:
- O = (c/2, 0)
- H = (0, h)

The distance d is:
\[ d = \sqrt{(0 - c/2)^2 + (h - 0)^2} \]
\[ d = \sqrt{(c/2)^2 + h^2} \]

### Step 6: Relate h to c
In a right-angled triangle, by the Pythagorean theorem, we know that:
\[ c^2 = a^2 + h^2 \]
where 'a' is the other leg of the triangle. Rearranging gives:
\[ h^2 = c^2 - a^2 \]

### Step 7: Substitute back into the distance formula
Substituting h^2 into the distance formula:
\[ d = \sqrt{(c/2)^2 + (c^2 - a^2)} \]
Since the hypotenuse c is the longest side, we can simplify this further.

### Conclusion
For a right-angled triangle, the distance between the circumcenter and orthocenter is half of the hypotenuse.

### Final Answer
The distance between the circumcenter and orthocenter of a right-angled triangle is **half the hypotenuse**.

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