In triangle PQR, a straight line is drawn parallel to the base QR and it cuts side PQ at point X and side PR at point Y. If the ratio PX : XQ is 5 : 6, then what is the ratio of the length XY to the length QR?

Introduction / Context:
This geometry question is based on similar triangles created by a line drawn parallel to one side of a triangle. When a line is parallel to the base of a triangle and cuts the other two sides, the smaller triangle formed near the vertex is similar to the original triangle. This similarity allows us to relate corresponding side lengths using simple ratios.

Given Data / Assumptions:

• Triangle PQR has base QR.
• A line parallel to QR meets side PQ at X and side PR at Y.
• The ratio PX : XQ is given as 5 : 6.
• We need to find the ratio XY : QR.
• All segments are straight and the triangle is non degenerate.

Concept / Approach:
If a line is drawn parallel to one side of a triangle and intersects the other two sides, then the smaller triangle formed is similar to the entire triangle by the basic proportionality theorem and similarity rules. The ratio of similarity is given by corresponding sides from the vertex to the cut point on each side. In this case, PX corresponds to PQ, and XY corresponds to QR. Therefore, XY / QR equals PX / PQ. We express PX / PQ in terms of the given ratio PX : XQ.

Step-by-Step Solution:
Given PX : XQ = 5 : 6. Let PX = 5k and XQ = 6k for some positive k. Then PQ = PX + XQ = 5k + 6k = 11k. The small triangle PXY is similar to triangle PQR, so the linear scale factor is PX / PQ. Thus, scale factor = PX / PQ = 5k / 11k = 5 / 11. Corresponding sides in similar triangles are proportional, so XY / QR = 5 / 11. Therefore, the ratio XY : QR = 5 : 11.

Verification / Alternative check:
We can check with assumed lengths. Suppose PQ = 11 units and PR is some length consistent with a general triangle. Take PX = 5 units and XQ = 6 units as per the ratio. Then triangle PXY is 5 / 11 scaled down compared to PQR. If QR is, say, 22 units, then XY should be 5 / 11 of 22, which is 10 units. The ratio XY : QR = 10 : 22 simplifies to 5 : 11, confirming the result. The specific length chosen for QR does not matter, only the proportionality does.

Why Other Options Are Wrong:
6 : 5 and 11 : 5 invert or otherwise distort the correct similarity ratio and would imply that the inner segment XY is longer than the entire base QR, which is impossible. 11 : 6 and 4 : 11 also do not follow from the similarity factor PX / PQ = 5 / 11 and are therefore inconsistent with the geometry of the figure.

Common Pitfalls:
A major source of error is to misinterpret PX : XQ as PX : PQ or to treat PX and XQ as directly proportional to XY and QR. Another pitfall is forgetting that PQ is the sum PX + XQ. Some students also mistakenly apply area ratios instead of length ratios. Carefully identifying which sides correspond in similar triangles and working step by step with the given ratio avoids these mistakes.

Final Answer:
The required ratio of XY to QR is 5 : 11.