Let G be the centroid of triangle ABC and the circumcircle of triangle AGC touches the side AB at A
If AC = 1, then the length of the median of triangle ABC through the vertex A is equal to

Options

A. `(sqrt3)/(2)`
B. `(1)/(2)`
C. `(2)/(sqrt3)`
D. `(5)/(sqrt2)`

Explanation

To solve the problem, we need to find the length of the median \( AD \) of triangle \( ABC \) through vertex \( A \) given that \( AC = 1 \) and \( G \) is the centroid of triangle \( ABC \).

### Step-by-Step Solution:

1. **Understanding the Triangle and Given Information:**
- We have triangle \( ABC \) with \( AC = 1 \).
- Let \( AB = b \) and \( BC = c \).
- The centroid \( G \) divides each median in the ratio \( 2:1 \).

2. **Using the Median Formula:**
- The length of the median \( AD \) from vertex \( A \) to side \( BC \) is given by the formula:
\[ AD = \sqrt{\frac{2b^2 + 2c^2 - a^2}{4}} \]
- Here, \( a = BC \), \( b = AC = 1 \), and \( c = AB \).

3. **Identifying the Triangle Properties:**
- Since \( G \) is the centroid and the circumcircle of triangle \( AGC \) touches side \( AB \) at \( A \), triangle \( ABC \) is likely equilateral.
- Therefore, we can assume \( AB = AC = BC = 1 \).

4. **Substituting Values into the Median Formula:**
- Substituting \( b = 1 \) and \( c = 1 \) into the median formula:
\[ AD = \sqrt{\frac{2(1^2) + 2(1^2) - (1^2)}{4}} \]
- Simplifying this:
\[ AD = \sqrt{\frac{2 + 2 - 1}{4}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

5. **Final Result:**
- The length of the median \( AD \) through vertex \( A \) is:
\[ AD = \frac{\sqrt{3}}{2} \]