If ABC is an equilateral triangle and D is a point on BC such that `AD_|_BC`, then

Options

A. `AB : BD = 1:1`
B. `AB : BD = 1:2`
C. `AB : BD = 2:1 `
D. `AB: BD = 3:2 `

Answer

To solve the problem, we need to find the ratio of \( AB \) to \( BD \) in the given equilateral triangle \( ABC \) with point \( D \) on \( BC \) such that \( AD \) is perpendicular to \( BC \).

### Step-by-Step Solution:

1. **Draw the Equilateral Triangle**:
- Start by drawing triangle \( ABC \) where \( AB = BC = CA \). Let's assume each side of the triangle is of length 1 unit for simplicity.

2. **Identify the Point D**:
- Since \( AD \) is perpendicular to \( BC \), point \( D \) lies on line segment \( BC \).

3. **Use the Property of Isosceles Triangles**:
- In an equilateral triangle, when a perpendicular is drawn from a vertex to the opposite side, it bisects that side. Hence, \( D \) is the midpoint of \( BC \).

4. **Calculate Lengths**:
- Since \( D \) is the midpoint of \( BC \), we have:
\[ BD = DC = \frac{BC}{2} = \frac{1}{2} \]

5. **Find the Ratio \( AB : BD \)**:
- We know \( AB = 1 \) (since it's an equilateral triangle).
- Now, we can find the ratio:
\[ \text{Ratio } AB : BD = \frac{AB}{BD} = \frac{1}{\frac{1}{2}} = 2 \]

6. **Final Answer**:
- Therefore, the ratio \( AB : BD \) is \( 2 : 1 \).