If D is such a point on the side BC of `triangle ABC` that AB/AC=BD/CD, then AD must be a/an :

Options

A. altitude of `triangleABC`
B. median of `triangleABC`
C. angle bisector of `triangleABC`
D. perpendicular bisector of `triangleABC`

Answer

To solve the problem, we need to determine what type of line segment AD is in triangle ABC, given the condition that \( \frac{AB}{AC} = \frac{BD}{CD} \).

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
We are given that \( \frac{AB}{AC} = \frac{BD}{CD} \). This means that the ratio of the lengths of sides AB and AC is equal to the ratio of the lengths of segments BD and CD.

2. **Applying the Angle Bisector Theorem**:
The Angle Bisector Theorem states that if a point D lies on the side BC of triangle ABC such that \( \frac{AB}{AC} = \frac{BD}{CD} \), then AD is the angle bisector of angle A. This theorem indicates that the point D divides the opposite side (BC) in the same ratio as the other two sides (AB and AC).

3. **Conclusion**:
Since the condition \( \frac{AB}{AC} = \frac{BD}{CD} \) holds true, we can conclude that AD must be the angle bisector of angle A in triangle ABC.

### Final Answer:
AD must be an **angle bisector** of triangle ABC.

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