Solution:

Given, in an equilateral triangle ABC,

- AD is perpendicular to BC
- AD^2 = x.BC^2

To find the value of x, we need to use the Pythagoras theorem and trigonometric ratios.

Step 1: Draw an equilateral triangle ABC with AD perpendicular to BC.

Step 2: Label the sides and angles of the triangle. Let the length of BC be a.

Step 3: Use the Pythagoras theorem to find the length of BD and DC.

- In triangle ABD, we have AB^2 = AD^2 + BD^2
- In triangle ACD, we have AC^2 = AD^2 + DC^2

Since the triangle ABC is equilateral, we have AB = AC = BC = a.

Substituting AB = AC = BC = a, we get:

a^2 = AD^2 + BD^2
a^2 = AD^2 + DC^2

Adding these two equations, we get:

2a^2 = 2AD^2 + BD^2 + DC^2

Since the triangle ABC is equilateral, we have BD = DC = a/2.

Substituting BD = DC = a/2, we get:

2a^2 = 2AD^2 + 2(a/2)^2

2a^2 = 2AD^2 + a^2/2

Simplifying, we get:

AD^2 = (2/3)a^2

Step 4: Use the given relation AD^2 = x.BC^2 to find the value of x.

Substituting AD^2 = (2/3)a^2 and BC^2 = a^2, we get:

(2/3)a^2 = x.a^2

Simplifying, we get:

x = 2/3

Therefore, the value of x is 2/3.

Final Answer: x = 2/3.