Triangle ABC is similar to triangle PQR and AB : PQ = 2 : 3. AD is the median to the side BC in triangle ABC and PS is the median to the side QR in triangle PQR. What is the value of `(BD//QS)^2`?

Options

A. `3//5`
B. `4//9`
C. `2/3`
D. `4//7`

Answer

To solve the problem, we need to find the value of \((BD/QS)^2\) given that triangle ABC is similar to triangle PQR and the ratio of their corresponding sides is \(AB : PQ = 2 : 3\).

### Step-by-step Solution:

1. **Understanding Similar Triangles**:
Since triangle ABC is similar to triangle PQR, the ratios of their corresponding sides are equal. This means:
\[ \frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR} = \frac{2}{3} \]

2. **Identifying Medians**:
In triangle ABC, AD is the median to side BC, and in triangle PQR, PS is the median to side QR. The medians divide the sides into two equal halves.

3. **Finding Ratios of Medians**:
Since the triangles are similar, the lengths of the medians will also be in the same ratio as the sides of the triangles. Therefore, we can say:
\[ \frac{BD}{QS} = \frac{AB}{PQ} = \frac{2}{3} \]

4. **Calculating \((BD/QS)^2\)**:
Now, we need to square the ratio we found:
\[ \left(\frac{BD}{QS}\right)^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]

5. **Final Answer**:
Thus, the value of \((BD/QS)^2\) is:
\[ \frac{4}{9} \]

### Conclusion:
The correct answer is \(\frac{4}{9}\), which corresponds to option 2.