`DeltaABC` is an equilateral triangle such that AD`bot` BC, then `AD^(2) =`

Options

A. `(3)/(2) DC^(2)`
B. `2DC^(2)`
C. `3CD^(2)`
D. `4DC^(2)`

Answer

To solve the problem, we need to find the square of AD in an equilateral triangle ABC where AD is perpendicular to BC.

### Step-by-Step Solution:

1. **Identify the Properties of the Equilateral Triangle**:
- In an equilateral triangle, all sides are equal. Let the length of each side (AB, BC, CA) be denoted as \( s \).
- Therefore, \( AB = BC = CA = s \).

2. **Divide the Base**:
- Since AD is perpendicular to BC, it divides BC into two equal segments. Let \( BD = DC = x \).
- Thus, \( BC = BD + DC = x + x = 2x \).
- From this, we can conclude that \( 2x = s \), which gives us \( x = \frac{s}{2} \).

3. **Apply the Pythagorean Theorem**:
- In triangle ADB, we can apply the Pythagorean theorem since it is a right triangle (with AD as the height).
- According to the theorem:
\[ AB^2 = AD^2 + BD^2 \]
- Substituting the known values:
\[ s^2 = AD^2 + \left(\frac{s}{2}\right)^2 \]

4. **Calculate \( BD^2 \)**:
- Calculate \( \left(\frac{s}{2}\right)^2 \):
\[ \left(\frac{s}{2}\right)^2 = \frac{s^2}{4} \]

5. **Substitute and Rearrange**:
- Substitute \( BD^2 \) into the equation:
\[ s^2 = AD^2 + \frac{s^2}{4} \]
- Rearranging gives:
\[ AD^2 = s^2 - \frac{s^2}{4} \]

6. **Combine Like Terms**:
- Combine the terms on the right:
\[ AD^2 = \frac{4s^2}{4} - \frac{s^2}{4} = \frac{3s^2}{4} \]

7. **Final Result**:
- Therefore, we have:
\[ AD^2 = \frac{3s^2}{4} \]

### Conclusion:
The value of \( AD^2 \) in the equilateral triangle ABC is \( \frac{3s^2}{4} \).