In a `DeltaABC, AB=AC and AD_|_BC`, then:

Options

A. `AB lt AD`
B. `AB gt AD`
C. `AB=AD`
D. `ABleAD`

Answer

To solve the problem step by step, we will analyze the given information and derive the relationship between the segments in triangle ABC.

### Step-by-Step Solution:

1. **Identify the Triangle**: We have triangle ABC where AB = AC, indicating that it is an isosceles triangle.

**Hint**: Recall that in an isosceles triangle, the two sides opposite the equal angles are equal.

2. **Draw the Perpendicular**: A perpendicular line AD is drawn from point A to line BC. This means that AD is the altitude of the triangle from vertex A to side BC.

**Hint**: Remember that the altitude of a triangle is a line segment from a vertex perpendicular to the opposite side.

3. **Properties of Isosceles Triangle**: Since triangle ABC is isosceles (AB = AC), the altitude AD also acts as the median to side BC. This means that point D, where AD meets BC, divides BC into two equal segments, BD and DC.

**Hint**: In an isosceles triangle, the altitude from the vertex to the base bisects the base.

4. **Analyze the Right Triangle**: In triangle ABD, we have a right triangle where AD is the altitude, AB is one of the equal sides, and BD is half of the base BC.

**Hint**: Use the Pythagorean theorem to analyze the relationship between the sides of the right triangle.

5. **Apply the Pythagorean Theorem**: According to the Pythagorean theorem:
\[ AB^2 = AD^2 + BD^2 \]
Since BD is a positive length, it follows that:
\[ AD^2 < AB^2 \]
Taking the square root of both sides gives:
\[ AD < AB \]

**Hint**: The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (AB) is equal to the sum of the squares of the other two sides (AD and BD).

6. **Conclusion**: Since we have established that AD < AB, we can conclude that AB is greater than AD. Therefore, the correct option is:
- AB > AD

**Hint**: Compare the lengths of the altitude and the sides of the triangle to determine their relationships.

### Final Answer:
AB is greater than AD (Option 2).