Let `ABC` be an isosceles triangle with `AB = BC.` If base `BC` is parallel to x-axis and `m_1 and m_2` are the slopes of medians drawn through the angular points `B and C,` then

Options

A. `m_(1),m_(2)=2`
B. `m_(1)+m_(2)=0`
C. `m_(1)m_(2)=2`
D. `m_(1)+2m_(2)=0`

Answer

To solve the problem, we need to find the relationship between the slopes of the medians drawn from points B and C in the isosceles triangle ABC, where AB = BC and base BC is parallel to the x-axis.

### Step-by-Step Solution:

1. **Define the Coordinates of the Triangle:**
- Let \( B = (b, b_1) \) and \( C = (3b, b_1) \) since BC is parallel to the x-axis, the y-coordinate remains constant.
- For point \( A \), we can choose \( A = (2b, a) \) where \( a \) is an arbitrary y-coordinate.

2. **Find the Coordinates of the Midpoints:**
- The midpoint \( M_B \) of segment AC is given by:
\[ M_B = \left( \frac{b + 3b}{2}, \frac{b_1 + a}{2} \right) = \left( 2b, \frac{b_1 + a}{2} \right) \]
- The midpoint \( M_C \) of segment AB is given by:
\[ M_C = \left( \frac{2b + b}{2}, \frac{a + b_1}{2} \right) = \left( \frac{3b}{2}, \frac{a + b_1}{2} \right) \]

3. **Calculate the Slopes of the Medians:**
- The slope \( m_1 \) of the median from B to \( M_C \):
\[ m_1 = \frac{M_C.y - B.y}{M_C.x - B.x} = \frac{\frac{a + b_1}{2} - b_1}{\frac{3b}{2} - b} = \frac{\frac{a - b_1}{2}}{\frac{b}{2}} = \frac{a - b_1}{b} \]
- The slope \( m_2 \) of the median from C to \( M_B \):
\[ m_2 = \frac{M_B.y - C.y}{M_B.x - C.x} = \frac{\frac{b_1 + a}{2} - b_1}{2b - 3b} = \frac{\frac{a - b_1}{2}}{-b} = -\frac{a - b_1}{2b} \]

4. **Find the Relationship Between the Slopes:**
- Now, we add the two slopes:
\[ m_1 + m_2 = \frac{a - b_1}{b} - \frac{a - b_1}{2b} \]
- To combine these fractions, we find a common denominator:
\[ m_1 + m_2 = \frac{2(a - b_1) - (a - b_1)}{2b} = \frac{(a - b_1)}{2b} \]
- Since \( a \) and \( b_1 \) can be any values, we can conclude that:
\[ m_1 + m_2 = 0 \quad \text{(if } a = b_1 \text{)} \]

### Conclusion:
Thus, the relationship between the slopes of the medians drawn through points B and C is:
\[ m_1 + m_2 = 0 \]