In triangle PQR, S lies on PQ and T lies on PR such that ST is parallel to QR. If S divides PQ in the ratio 4 : 1 and QR = 15 cm, what is the length of segment ST?

Introduction / Context:
This geometry question involves similar triangles formed when a line segment is drawn parallel to one side of a triangle. It tests your understanding of the basic proportionality theorem (also known as Thales theorem) and how lengths scale in similar triangles.

Given Data / Assumptions:

• Triangle PQR has S on PQ and T on PR.
• ST is parallel to QR.
• S divides PQ in the ratio 4 : 1, meaning PS : SQ = 4 : 1.
• QR, the base of the larger triangle, is 15 cm.
• We need to find the length ST, the corresponding side in the smaller similar triangle.

Concept / Approach:
Main ideas:

• If a line segment is drawn parallel to one side of a triangle and intersects the other two sides, it creates a smaller triangle similar to the original.
• The ratio of corresponding sides in similar triangles equals the ratio of any pair of corresponding segments on the intersected sides.
• Here, triangle PST is similar to triangle PQR.
• The scale factor is PS / PQ.

Step-by-Step Solution:
Given PS : SQ = 4 : 1. Then PQ = PS + SQ, so PQ is in parts 4 + 1 = 5. Thus PS / PQ = 4 / 5. Triangles PST and PQR are similar since ST ∥ QR. Therefore ST / QR = PS / PQ = 4 / 5. Given QR = 15 cm, we get ST = (4 / 5) * 15. Compute ST: (4 / 5) * 15 = 12. So ST = 12 cm.

Verification / Alternative check:
You can also think in terms of area scaling or other corresponding sides. Any side of the smaller triangle is 4/5 of the corresponding side of the larger triangle. If the side ratio is 4/5, then the base ST must be 4/5 of 15 cm, which is 12 cm, matching our calculation.

Why Other Options Are Wrong:
Option a: 3 cm is much smaller and would correspond to a scale factor of 1/5, not 4/5.
Option b: 5 cm does not relate to the 4 : 1 ratio and gives an incorrect scale factor.
Option d: 10 cm would correspond to a factor 2/3, which is not derived from the given ratio 4 : 1.
Option e: 7 cm has no simple proportional relationship with 15 cm and does not follow from the 4 : 1 division of PQ.

Common Pitfalls:
A common mistake is misinterpreting the ratio 4 : 1 as PS : PQ instead of PS : SQ, or forgetting to add the parts to find the whole PQ. Another error is to invert the ratio when relating ST to QR. Always keep track of which segment corresponds to which side in the similar triangles.

Final Answer:
The length of ST is 12 cm.