PQR is an equilateral triangle. A line segment MN is drawn parallel to side QR such that point M lies on side PQ and point N lies on side PR. If PN = 6 centimetres, then find the length of MN in centimetres.

Introduction / Context:
This geometry question is based on properties of equilateral triangles and parallels. It explores the relationship between a smaller triangle formed by drawing a segment parallel to one side and the original equilateral triangle. Recognising that the small triangle is also equilateral plays a key role in solving this problem quickly and correctly.

Given Data / Assumptions:

• PQR is an equilateral triangle, so all its sides are equal and all its angles are 60 degrees.
• Segment MN is drawn parallel to side QR.
• M lies on side PQ and N lies on side PR.
• PN is a side of the smaller triangle formed, and PN = 6 cm.
• We must find the length of segment MN in centimetres.
• Standard Euclidean geometry is assumed.

Concept / Approach:
Because PQR is equilateral (all angles 60 degrees) and MN is parallel to QR, triangle PMN inherits the same angle measures as triangle PQR. Specifically, angle at P is common, and parallels ensure corresponding angles at M and N match those at Q and R respectively. Therefore triangle PMN is similar to triangle PQR. In fact, with both triangles having all angles 60 degrees, triangle PMN is itself an equilateral triangle. That means all sides of triangle PMN are equal, including MN and PN.

Step-by-Step Solution:
Step 1: Note that triangle PQR is equilateral, so each angle is 60 degrees. Step 2: Segment MN is drawn parallel to QR with M on PQ and N on PR. Step 3: Because MN is parallel to QR, angle PMN equals angle PQR and angle PNM equals angle PRQ by corresponding angles. Step 4: Angle at P is common to both triangle PMN and triangle PQR. Step 5: Hence triangle PMN has all three angles equal to 60 degrees. Step 6: Any triangle with all angles 60 degrees is equilateral. Step 7: Therefore, in triangle PMN, all sides are equal: PM = MN = PN. Step 8: We are given PN = 6 cm, so MN = 6 cm.

Verification / Alternative check:
We can also reason in terms of similarity ratios. Since triangle PMN is similar to triangle PQR with all angles equal, the scale factor is the same for each side. Side PN is corresponding to side PR, and side MN corresponds to QR. But within triangle PMN itself, each side must have the same length. The given single side PN = 6 cm fixes the scale factor, which then forces MN to equal 6 cm as well. Any different value for MN would break the equilateral property.

Why Other Options Are Wrong:
3 cm, 4.5 cm, 9 cm and 12 cm: These values would make triangle PMN non-equilateral if PN remains 6 cm, contradicting the angle-based similarity to triangle PQR and the parallel condition between MN and QR.

Common Pitfalls:
A common mistake is to assume some arbitrary ratio of similarity without first noticing that both triangles are equilateral. Some students may also try to bring in lengths of PQR that are not given or necessary. The key is to focus on the angles: an equilateral triangle with a line parallel to the base through points on the other two sides still creates a smaller equilateral triangle at the top, so all its sides must be equal.

Final Answer:
The length of segment MN is 6 cm.