If the medians PT and RS of a triangle with vertices P (0, b), Q (0, 0) and R (a, 0) are perpendicular to each other, which of the following satisfies the relationship between a and b?

Options

A. `4b^(2) = a^(2)`
B. `2b^(2) = a^(2)`
C. `a = - 2b`
D. `a^(2) + b^(2) = 0`

Answer

To solve the problem, we need to find the relationship between \( a \) and \( b \) given that the medians \( PT \) and \( RS \) of triangle \( PQR \) are perpendicular to each other. The vertices of the triangle are given as \( P(0, b) \), \( Q(0, 0) \), and \( R(a, 0) \).

### Step-by-step Solution:

1. **Identify the Midpoints**:
- The median \( PT \) is from vertex \( P \) to the midpoint \( T \) of side \( QR \).
- The coordinates of \( T \) (midpoint of \( QR \)) can be calculated as:
\[ T = \left( \frac{0 + a}{2}, \frac{0 + 0}{2} \right) = \left( \frac{a}{2}, 0 \right) \]

- The median \( RS \) is from vertex \( R \) to the midpoint \( S \) of side \( PQ \).
- The coordinates of \( S \) (midpoint of \( PQ \)) can be calculated as:
\[ S = \left( \frac{0 + 0}{2}, \frac{b + 0}{2} \right) = \left( 0, \frac{b}{2} \right) \]

2. **Calculate the Slopes of the Medians**:
- The slope \( m_1 \) of median \( PT \) can be calculated using the coordinates of \( P(0, b) \) and \( T\left(\frac{a}{2}, 0\right) \):
\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - b}{\frac{a}{2} - 0} = \frac{-b}{\frac{a}{2}} = -\frac{2b}{a} \]

- The slope \( m_2 \) of median \( RS \) can be calculated using the coordinates of \( R(a, 0) \) and \( S(0, \frac{b}{2}) \):
\[ m_2 = \frac{\frac{b}{2} - 0}{0 - a} = \frac{\frac{b}{2}}{-a} = -\frac{b}{2a} \]

3. **Set Up the Perpendicularity Condition**:
- Since the medians \( PT \) and \( RS \) are perpendicular, the product of their slopes should equal \(-1\):
\[ m_1 \cdot m_2 = -1 \]
- Substituting the values of \( m_1 \) and \( m_2 \):
\[ \left(-\frac{2b}{a}\right) \cdot \left(-\frac{b}{2a}\right) = -1 \]
- Simplifying gives:
\[ \frac{2b^2}{2a^2} = -1 \implies \frac{b^2}{a^2} = -1 \]

4. **Final Relationship**:
- Since \( b^2 \) and \( a^2 \) are both non-negative, the only way for this equation to hold true is if both \( a \) and \( b \) are zero:
\[ a^2 + b^2 = 0 \]
- This implies:
\[ a = 0 \quad \text{and} \quad b = 0 \]

### Conclusion:
The relationship between \( a \) and \( b \) is given by:
\[ a^2 + b^2 = 0 \]