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

Introduction / Context:
This problem combines the concepts of similar triangles and medians. It tests whether you understand that when triangles are similar, not only their sides but also certain segments such as medians and altitudes are in the same ratio. You must then use that information to compute a squared ratio involving half segments of corresponding sides.

Given Data / Assumptions:

• Triangle ABC is similar to triangle PQR.
• The ratio of corresponding sides is AB : PQ = 2 : 3.
• AD is a median to side BC in triangle ABC, so D is the midpoint of BC.
• PS is a median to side QR in triangle PQR, so S is the midpoint of QR.
• We must find the value of (BD / QS)^2.
• Triangles are assumed to be in a Euclidean plane and similarity is strict.

Concept / Approach:
When two triangles are similar, the ratio of any pair of corresponding linear measures is constant. This applies to sides, medians, and altitudes. Because AD and PS are medians, their lengths are proportional to the corresponding sides: AD / PS = AB / PQ. Also, a median divides the opposite side into two equal segments, so BD is half of BC and QS is half of QR. Since BC and QR are corresponding sides of similar triangles, BC / QR = AB / PQ. Therefore BD / QS has the same ratio as BC / QR.

Step-by-Step Solution:
Step 1: From similarity, AB / PQ = 2 / 3. Step 2: For similar triangles, the ratio of corresponding sides is constant, so BC / QR = 2 / 3 as well. Step 3: Because AD is a median, BD = BC / 2. Step 4: Because PS is a median, QS = QR / 2. Step 5: Compute BD / QS: BD / QS = (BC / 2) / (QR / 2) = BC / QR. Step 6: So BD / QS = BC / QR = 2 / 3. Step 7: Now compute (BD / QS)^2 = (2 / 3)^2 = 4 / 9. Step 8: Thus, the required value is 4 / 9.

Verification / Alternative check:
We can verify by thinking numerically. Suppose AB = 2 units and PQ = 3 units. Then take BC = 2k and QR = 3k for some positive k. Then BD = BC / 2 = k and QS = QR / 2 = 1.5k. Therefore BD / QS = k / (1.5k) = 2 / 3. Squaring gives (BD / QS)^2 = 4 / 9, which matches the analytic reasoning and is independent of the particular value of k.

Why Other Options Are Wrong:
The value 3 / 5 does not arise from the square of the 2 / 3 ratio. The option 2 / 3 is the ratio BD / QS, not its square. The fraction 4 / 7 is unrelated to the given side ratio 2 : 3. The option 5 / 9 also does not correspond to squaring 2 / 3 or any direct consequence of the given similarity ratio. Only 4 / 9 correctly matches (2 / 3)^2.

Common Pitfalls:
A frequent error is to confuse BD / QS with AB / PQ itself and forget to square the ratio at the end. Some students also mix up which segments are halves of which sides and mistakenly think BD and QS are in a different ratio. Others attempt to introduce unnecessary variables for the medians, which complicates the problem. Remembering that medians to corresponding sides scale in the same way as the sides themselves and that halving both sides cancels out in the ratio is crucial.

Final Answer:
The value of (BD / QS)^2 is 4/9.