In triangle PQR, points L and M lie on sides PQ and PR respectively such that segment LM is parallel to side QR. If PL = 2 cm, LQ = 6 cm and PM = 1.5 cm, then find the length of segment MR in centimetres.

Introduction / Context:
This geometry question tests your understanding of similar triangles and the basic proportionality theorem (also known as Thales theorem). When a line segment is drawn parallel to one side of a triangle, it creates a smaller triangle inside the original triangle that is similar to it, and corresponding sides are in proportion. Using these proportional relationships, we can find unknown segment lengths such as MR in this problem.

Given Data / Assumptions:

• Triangle PQR is given.
• L lies on side PQ and M lies on side PR.
• LM is parallel to QR.
• PL = 2 cm and LQ = 6 cm, so PQ = PL + LQ = 8 cm.
• PM = 1.5 cm.
• We are required to find MR in centimetres.
• All geometry is in a standard Euclidean plane.

Concept / Approach:
When a line segment joins two sides of a triangle and is parallel to the third side, the smaller triangle formed is similar to the original triangle. Here triangle PLM is similar to triangle PQR. Therefore, the ratios of corresponding sides are equal. In particular, PL / PQ = PM / PR. Once we compute PR, we get MR by subtracting PM from the total length PR since PR = PM + MR.

Step-by-Step Solution:
Step 1: Compute the full length of side PQ. PQ = PL + LQ = 2 + 6 = 8 cm. Step 2: Because LM is parallel to QR, triangles PLM and PQR are similar. Step 3: Write the similarity ratio: PL / PQ = PM / PR. Step 4: Substitute known values: 2 / 8 = 1.5 / PR. Step 5: Simplify the left side: 2 / 8 = 1 / 4. Step 6: So 1 / 4 = 1.5 / PR, hence PR = 1.5 * 4 = 6 cm. Step 7: Segment PR consists of PM and MR. Thus PR = PM + MR. Step 8: Substitute values: 6 = 1.5 + MR, so MR = 6 - 1.5 = 4.5 cm.

Verification / Alternative check:
We can check the proportionality again using PR = 6 cm and MR = 4.5 cm. Since PM = 1.5 cm, the ratio PM : PR = 1.5 : 6 = 1 : 4, which matches PL : PQ = 2 : 8 = 1 : 4. This confirms that the similarity ratio is satisfied and that the computed PR and MR are consistent with the given data and the parallel line condition.

Why Other Options Are Wrong:
0.5 cm: This would make PR = 2.0 cm, contradicting the proportionality PL / PQ = PM / PR.
8 cm and 9 cm: These imply much larger values of PR and do not satisfy the similarity ratio between PL / PQ and PM / PR.
6 cm: This is the value of PR itself, not MR. Using MR = 6 cm would make PR = 7.5 cm, which again breaks the proportional relationship.

Common Pitfalls:
A common mistake is to assume that the ratio PL : LQ equals PM : MR directly without using the full sides. Another frequent error is to forget that PR is the sum of PM and MR, not just one of them. Some learners also mix up which sides correspond in the similar triangles. Always map corresponding vertices carefully and then apply the proportionality to whole sides before subtracting to obtain internal segments.

Final Answer:
The length of segment MR is 4.5 cm.