In triangle PQR, points S and T lie on sides PQ and PR respectively such that segment ST is parallel to QR. If PS : SQ = 3 : 5 and PR = 6 cm, then what is the length of segment PT (in centimetres)?

Introduction / Context:
This question uses the basic concept of similar triangles created by a line drawn parallel to one side of a triangle. Aptitude tests frequently use such problems to check understanding of proportionality of corresponding sides and how segment ratios give actual lengths.

Given Data / Assumptions:
- Triangle PQR is given.
- Point S lies on side PQ and point T lies on side PR.
- Segment ST is parallel to base QR.
- The ratio PS : SQ is 3 : 5.
- The full side PR has length 6 cm.
- We need the length of PT.

Concept / Approach:
When a line is drawn parallel to the base of a triangle and cuts the other two sides, a smaller triangle similar to the original triangle is formed. Here, triangle PST is similar to triangle PQR. For similar triangles, the ratio of corresponding sides is equal. The segment PS corresponds to PQ, and PT corresponds to PR, so the ratio PS / PQ equals PT / PR.

Step-by-Step Solution:
Step 1: From PS : SQ = 3 : 5, write PS = 3k and SQ = 5k for some positive k. Step 2: Then the full side PQ = PS + SQ = 3k + 5k = 8k. Step 3: For similar triangles PST and PQR, PS / PQ = PT / PR. Step 4: Substitute PS = 3k and PQ = 8k to get (3k) / (8k) = PT / PR. Step 5: The k terms cancel, so 3 / 8 = PT / PR. Step 6: Given PR = 6 cm, we have PT = (3 / 8) * 6 = 18 / 8 = 2.25 cm.

Verification / Alternative Check:
As a check, note that PS / PQ = 3 / 8. This means the similar smaller triangle PST is a 3/8 scaled copy of PQR. Any length in triangle PST must be 3/8 of the corresponding length in triangle PQR. Since PT corresponds to PR, and PR = 6 cm, multiplying 6 by 3/8 again gives 2.25 cm, which confirms the previous calculation.

Why Other Options Are Wrong:
Option a, 2 cm, corresponds to a factor of 2/6 = 1/3, which does not match the required ratio 3/8.
Option c, 3.5 cm, is more than half of PR and conflicts with PS being only 3/8 of PQ in the similarity ratio.
Option d, 4 cm, would correspond to PT / PR = 4 / 6 = 2 / 3, again inconsistent with the ratio 3 / 8.

Common Pitfalls:
Learners sometimes mistakenly take PS : SQ as PS : PQ and then use that directly in place of PS / PQ. Another common mistake is to assume PS / SQ equals PT / PR, which is not a correct correspondence between the similar triangles. Drawing the diagram carefully and labeling corresponding sides in both triangles helps to avoid these errors.

Final Answer:
The length of PT is 2.25 cm.