In triangle PQR, points S and T lie on sides PQ and PR respectively. Segment ST is drawn parallel to side QR. If PS = 6 cm, SQ = 9 cm (so that PQ = PS + SQ), and PR = 12.5 cm, what is the length of segment TR?

Introduction / Context:
This question uses the concept of similar triangles formed by a line drawn parallel to one side of a triangle. In triangle PQR, a segment ST is drawn parallel to base QR, cutting the other two sides PQ and PR at points S and T respectively. We are given PS, SQ and PR, and we are asked to find the remaining part TR of side PR. This is a standard application of basic proportionality theorem, also known as Thales theorem, in plane geometry.

Given Data / Assumptions:
- Triangle PQR is a general triangle. - Point S lies on side PQ, and point T lies on side PR. - ST is parallel to QR. - PS = 6 cm and SQ = 9 cm, so PQ = PS + SQ = 15 cm. - PR = 12.5 cm. - All lengths are positive and measured in centimetres.

Concept / Approach:
When a line segment is drawn parallel to one side of a triangle and meets the other two sides, it divides those sides proportionally. Because ST is parallel to QR, triangle PST is similar to triangle PQR. Therefore, the ratio PS / PQ equals PT / PR. Once we know PS and PQ, we can find PT using proportionality, and then TR is simply PR minus PT. This avoids any need for angles or coordinate geometry and uses only ratio reasoning.

Step-by-Step Solution:
Step 1: From PS = 6 cm and SQ = 9 cm, compute PQ. Step 2: PQ = PS + SQ = 6 + 9 = 15 cm. Step 3: Because ST is parallel to QR, triangles PST and PQR are similar. Step 4: By similarity, corresponding sides are proportional. So PS / PQ = PT / PR. Step 5: Substitute known values: 6 / 15 = PT / 12.5. Step 6: Simplify 6 / 15 to 2 / 5. Step 7: So 2 / 5 = PT / 12.5, hence PT = (2 / 5) * 12.5. Step 8: Compute PT = 25 / 5 = 5 cm. Step 9: Now PR = 12.5 cm, and PR is made up of PT + TR. Step 10: Therefore TR = PR − PT = 12.5 − 5 = 7.5 cm.

Verification / Alternative check:
We can also check by computing the scale factor between triangles PST and PQR. The ratio PS : PQ is 6 : 15 which equals 2 : 5. Thus every corresponding length in the smaller triangle PST is 2 / 5 of the length in the larger triangle PQR. Since PR is 12.5 cm, the corresponding side PT must be (2 / 5) * 12.5 = 5 cm, which matches our earlier calculation. Then TR is the remaining 7.5 cm. The consistency of both methods confirms the answer.

Why Other Options Are Wrong:
Option 5 cm equals PT, not TR, so it represents the upper part of PR instead of the required lower segment. Option 10 cm would make PT equal to 2.5 cm, which does not preserve the similarity ratio 2 : 5 between PS and PQ. Option 2.5 cm would make PT equal to 10 cm, again reversing the proportionality and contradicting the condition that PS : PQ = PT : PR.

Common Pitfalls:
Learners sometimes directly apply ratios to TR instead of PT, or they forget that PS and PQ must be added to get the full side length. Another mistake is mixing up the direction of the ratio, such as taking PQ / PS instead of PS / PQ when writing PT / PR. Always match smaller triangle sides to the corresponding sides in the larger triangle in the same order to avoid proportionality errors.

Final Answer:
The length of segment TR is 7.5 cm.