If hypotenuse of a right angle is 15 cm then wht is the distance between its orthocentre and centroid ?

Options

A. 5 cm
B. 10 cm
C. `10/3` cm
D. `20/3` cm

Answer

To solve the problem of finding the distance between the orthocenter and centroid of a right triangle with a hypotenuse of 15 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Triangle**: We have a right triangle with a hypotenuse (AC) of 15 cm. Let's denote the vertices of the triangle as A, B, and C, where angle B is the right angle.

2. **Find the Length of the Median**: The median (BD) from vertex B to the hypotenuse (AC) divides the hypotenuse into two equal parts. Since the hypotenuse AC is 15 cm, each segment (AD and DC) will be:
\[ AD = DC = \frac{15}{2} = 7.5 \text{ cm} \]

3. **Use the Median Length Formula**: The length of the median (BD) in a right triangle can be calculated using the formula:
\[ BD = \frac{1}{2} \sqrt{2AB^2 + 2BC^2 - AC^2} \]
However, for a right triangle, we can directly say that the length of the median to the hypotenuse is:
\[ BD = \frac{AC}{2} = \frac{15}{2} = 7.5 \text{ cm} \]

4. **Determine the Position of the Centroid (G)**: The centroid (G) divides the median (BD) in the ratio 2:1. This means:
- Let BG = 2x and GD = x.
- Therefore, \( BG + GD = BD \) leads to:
\[ 2x + x = 7.5 \implies 3x = 7.5 \implies x = \frac{7.5}{3} = 2.5 \text{ cm} \]

5. **Calculate BG**: Since \( BG = 2x \):
\[ BG = 2 \times 2.5 = 5 \text{ cm} \]

6. **Conclusion**: The distance between the orthocenter (B) and the centroid (G) is:
\[ \text{Distance} = BG = 5 \text{ cm} \]

### Final Answer:
The distance between the orthocenter and centroid of the triangle is **5 cm**.

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