`DeltaABC` a right angled triangle has `/_B = 90^@` and `AC `is hypotenuse. D is its circumcentre and `AB=3 cm, BC = 4 cm`. The value of `BD` is

Options

A. 3 cm.
B. 4 cm.
C. 2.5 cm.
D. 5.5 cm.

Answer

To solve the problem step by step, we will follow the necessary geometric principles and calculations.

### Step-by-Step Solution:

1. **Identify the Triangle and Given Information:**
- We have a right-angled triangle \( \Delta ABC \) with \( \angle B = 90^\circ \).
- The lengths of the sides are given as \( AB = 3 \, \text{cm} \) and \( BC = 4 \, \text{cm} \).
- The hypotenuse \( AC \) is opposite the right angle.

**Hint:** Draw a diagram of triangle \( ABC \) to visualize the problem.

2. **Apply the Pythagorean Theorem:**
- According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
- Therefore, we can write:
\[ AC^2 = AB^2 + BC^2 \]

3. **Substitute the Known Values:**
- Substitute \( AB = 3 \, \text{cm} \) and \( BC = 4 \, \text{cm} \) into the equation:
\[ AC^2 = 3^2 + 4^2 \]
\[ AC^2 = 9 + 16 = 25 \]

**Hint:** Remember that squaring a number means multiplying it by itself.

4. **Calculate the Length of the Hypotenuse:**
- To find \( AC \), take the square root of both sides:
\[ AC = \sqrt{25} = 5 \, \text{cm} \]

**Hint:** The square root of a number gives you the original number that was squared.

5. **Determine the Circumcenter:**
- The circumcenter \( D \) of a triangle is the point where the perpendicular bisectors of the sides intersect. In a right triangle, the circumcenter is located at the midpoint of the hypotenuse.
- Since \( AC = 5 \, \text{cm} \), the distance from \( D \) to each vertex \( A \), \( B \), and \( C \) is equal to half the length of the hypotenuse:
\[ AD = CD = BD = \frac{AC}{2} = \frac{5}{2} = 2.5 \, \text{cm} \]

**Hint:** The circumcenter is equidistant from all three vertices of the triangle.

6. **Conclusion:**
- Therefore, the value of \( BD \) is \( 2.5 \, \text{cm} \).

### Final Answer:
The value of \( BD \) is \( 2.5 \, \text{cm} \).