In a `triangle ABC`,AB = AC and `AD bot BC`, then :

Options

A. `ABltAD`
B. `ABgtAD`
C. AB=AD
D. `ABleAD`

Answer

To solve the problem step by step, we will analyze the triangle ABC where AB = AC and AD is perpendicular to BC.

### Step 1: Understand the Triangle
We have triangle ABC where:
- AB = AC (This means triangle ABC is isosceles)
- AD is perpendicular to BC (This means that AD forms a right angle with BC)

### Step 2: Identify the Points
Let:
- A be the vertex opposite to the base BC.
- B and C be the other two vertices.
- D be the foot of the perpendicular from A to BC.

### Step 3: Analyze the Isosceles Triangle
Since AB = AC, the angles opposite these sides will also be equal. Therefore:
- Let angle ABC = angle ACB = x (for some angle x).
- The angle at A (angle BAC) will then be 180° - 2x.

### Step 4: Use Right Triangle Properties
In right triangle ABD:
- AD is the height from A to BC.
- By the properties of triangles, we can relate the sides using trigonometric ratios.

### Step 5: Compare the Lengths
We need to compare the lengths of AB and AD:
- In any triangle, the length of the side opposite an angle is greater than the length of the altitude from that angle to the opposite side.
- Thus, since AB is opposite angle A, and AD is the altitude, we can conclude that:
\[ AB > AD \]

### Conclusion
From the analysis, we find that the length of side AB is greater than the length of altitude AD.

### Final Answer
AB > AD