a and b are two side adjacent to the right angle of a right-angled triangle and p is the perpendicular drawn to the hypotenuse from the opposite vertex. Then `p^(2)` is equal to

Options

A. `a^(2) + b^(2)`
B. `(1)/(a^(2)) + (1)/(b^(2))`
C. `(a^(2)b^(2))/(a^(2)+b^(2))`
D. `a^(2) - b^(2)`

Answer

To solve the problem, we need to find the value of \( p^2 \) where \( p \) is the perpendicular drawn from the opposite vertex to the hypotenuse of a right-angled triangle with sides \( a \) and \( b \).

### Step-by-Step Solution:

1. **Identify the Triangle and its Components**:
- Let \( \triangle ABC \) be a right-angled triangle where \( \angle ABC = 90^\circ \).
- Let \( AB = c \) (the hypotenuse), \( BC = a \), and \( AC = b \).
- The point \( P \) is the foot of the perpendicular drawn from vertex \( A \) to the hypotenuse \( BC \).

2. **Area of Triangle Using Base and Height**:
- The area of triangle \( ABC \) can be calculated using the two sides adjacent to the right angle:
\[ \text{Area} = \frac{1}{2} \times a \times b \]

3. **Area of Triangle Using Hypotenuse and Height**:
- The area can also be expressed using the hypotenuse \( c \) and the height \( p \):
\[ \text{Area} = \frac{1}{2} \times c \times p \]

4. **Equating the Two Area Expressions**:
- Since both expressions represent the same area, we can set them equal to each other:
\[ \frac{1}{2} \times a \times b = \frac{1}{2} \times c \times p \]
- Simplifying this gives:
\[ ab = cp \quad \text{(Equation 1)} \]

5. **Using Pythagorean Theorem**:
- According to the Pythagorean theorem:
\[ c^2 = a^2 + b^2 \]
- Therefore, we can express \( c \) as:
\[ c = \sqrt{a^2 + b^2} \]

6. **Substituting \( c \) into Equation 1**:
- Substitute \( c \) into Equation 1:
\[ ab = \sqrt{a^2 + b^2} \cdot p \]

7. **Solving for \( p \)**:
- Rearranging gives:
\[ p = \frac{ab}{\sqrt{a^2 + b^2}} \]

8. **Finding \( p^2 \)**:
- Now, squaring both sides to find \( p^2 \):
\[ p^2 = \left(\frac{ab}{\sqrt{a^2 + b^2}}\right)^2 \]
- This simplifies to:
\[ p^2 = \frac{a^2b^2}{a^2 + b^2} \]

### Conclusion:
Thus, the value of \( p^2 \) is:
\[ p^2 = \frac{a^2b^2}{a^2 + b^2} \]