Given:
Triangle ABC is an equilateral triangle.
AD is perpendicular to BC.

To find:
The value of AD^2.

Solution:
Let's analyze the given problem step by step.

Step 1: Understanding the given information:
We are given an equilateral triangle ABC with AD perpendicular to BC. This means that AD is the altitude of the triangle.

Step 2: Properties of an equilateral triangle:
In an equilateral triangle, all sides are equal, and all angles are 60 degrees.
Let's denote the length of each side as "s".

Step 3: Analyzing the right-angled triangle ADB:
In triangle ADB, AD is the altitude, and AB is the base.
We can use the Pythagorean theorem to relate the sides of this right-angled triangle.

According to the Pythagorean theorem:
AB^2 = AD^2 + BD^2

Since triangle ABC is an equilateral triangle, AB = BC = AC = s.

Step 4: Finding the length of BD:
To find the length of BD, we need to divide the base side, AB, into two equal parts.
Since triangle ABC is an equilateral triangle, the base side AB is divided into two equal parts at point D.
Therefore, BD = AB/2 = s/2.

Step 5: Substituting the values into the Pythagorean theorem equation:
Now, we can substitute the values into the Pythagorean theorem equation for triangle ADB.

AB^2 = AD^2 + BD^2
s^2 = AD^2 + (s/2)^2
s^2 = AD^2 + s^2/4

Step 6: Simplifying the equation:
Let's simplify the equation obtained in the previous step.

s^2 - s^2/4 = AD^2
(4s^2 - s^2)/4 = AD^2
(3s^2)/4 = AD^2

Step 7: Finding the value of AD^2:
Finally, let's calculate the value of AD^2.

AD^2 = (3s^2)/4

Conclusion:
The value of AD^2 in terms of the side length s of the equilateral triangle ABC is (3s^2)/4.