Similar triangles with perimeter ratio 1:4:\nIf ΔGHI ~ ΔKLM and Perimeter(GHI) : Perimeter(KLM) = 1 : 4 with GH = 2 cm, find the exact length (in cm) of the corresponding side KL.

Introduction / Context:
For similar triangles, any linear dimension (including a side) scales in the same ratio as the perimeters. Hence, the side ratio equals the perimeter ratio.

Given Data / Assumptions:

• ΔGHI ~ ΔKLM.
• Perimeter ratio = 1 : 4 → linear scale = 1 : 4.
• GH corresponds to KL; GH = 2 cm.

Concept / Approach:
If k is the scale factor from GHI to KLM, then k = 4 (since perimeters scale by k). Therefore KL = k * GH.

Step-by-Step Solution:

Verification / Alternative check:
Any other corresponding side would be 4 times its match in ΔGHI; 8 cm is consistent with the 1:4 scale.

Why Other Options Are Wrong:
4 cm or 2 cm ignore the 4× scale; 16 cm or 32 cm over-scale.

Common Pitfalls:
Using area or volume scaling instead of linear/perimeter scaling.

Final Answer:
8 cm