If median of the ΔABC through A is perpendicular to BC. then which oen of the following is correct?

Options

A. `tanA+tanB=0`
B. `tanB-tanC=0`
C. `tanC+2tanA=0`
D. `tanB+tanC=0`

Answer

To solve the problem, we need to analyze the given information about triangle ABC, where the median from vertex A to side BC is perpendicular to BC. Let's go through the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Triangle**:
We have triangle ABC with vertices A, B, and C. The median from vertex A to side BC is drawn, and we denote the point where the median intersects BC as point D.

2. **Properties of the Median**:
Since D is the midpoint of BC (because D is where the median intersects BC), we have:
\[ BD = DC \]

3. **Perpendicularity**:
The problem states that the median AD is perpendicular to BC. Therefore, we have:
\[ \angle ADB = 90^\circ \quad \text{and} \quad \angle ADC = 90^\circ \]

4. **Congruent Triangles**:
We can now consider triangles ADB and ADC. Since AD is common to both triangles, and we have:
- \(BD = DC\) (as D is the midpoint),
- \(\angle ADB = \angle ADC = 90^\circ\),
we can apply the Side-Angle-Side (SAS) congruence criterion. Thus, we conclude:
\[ \triangle ADB \cong \triangle ADC \]

5. **Angles in Congruent Triangles**:
From the congruence of triangles ADB and ADC, we can deduce that:
\[ \angle ADB = \angle ADC \]
This implies that:
\[ \angle B = \angle C \]

6. **Conclusion**:
Since angles B and C are equal, we can express this in terms of tangent:
\[ \tan B = \tan C \]
Therefore, we can conclude:
\[ \tan B - \tan C = 0 \]
Hence, the correct option is option B.

### Final Answer:
The correct statement is that \( \tan B - \tan C = 0 \).

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