In a `Delta ABC, D` is a point on BC. AB is the hypotenus of then which relation is correct .

Options

A. `AB^(2)+CD^(2)=BC^(2)+AD^(2)`
B. `CD^(2)+BD^(2)=2AD^(2)`
C. `AB^(2)+AC^(2)=2AD^(2)`
D. `AB^(2)+AD^(2)=BD^(2)`

Answer

To solve the problem, we need to analyze the triangle ABC with point D on side BC, where AB is the hypotenuse. We will use the Pythagorean theorem to derive the relationships between the sides of the triangles involved.

### Step-by-Step Solution:

1. **Identify the Right Triangle**:
- We have triangle ABC where AB is the hypotenuse.
- By definition, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

2. **Apply the Pythagorean Theorem to Triangle ABC**:
- According to the Pythagorean theorem:
\[ AB^2 = AC^2 + BC^2 \tag{1} \]

3. **Consider Triangle ACD**:
- Triangle ACD is also a right triangle (assuming D is on BC).
- Applying the Pythagorean theorem here gives:
\[ AD^2 = AC^2 + CD^2 \tag{2} \]

4. **Rearrange Equation (2)**:
- From equation (2), we can express \(AC^2\) in terms of \(AD^2\) and \(CD^2\):
\[ AC^2 = AD^2 - CD^2 \tag{3} \]

5. **Substitute Equation (3) into Equation (1)**:
- Now, substitute \(AC^2\) from equation (3) into equation (1):
\[ AB^2 = (AD^2 - CD^2) + BC^2 \]
- Rearranging this gives:
\[ AB^2 + CD^2 = AD^2 + BC^2 \tag{4} \]

6. **Conclusion**:
- The correct relation derived from the above steps is:
\[ AB^2 + CD^2 = BC^2 + AD^2 \]
- Thus, the correct option is option number 1.