In triangle DEF, points G and H lie on sides DE and DF respectively, and GH is parallel to EF. Point G divides side DE in the ratio DG : GE = 3 : 2. If HF = 8 cm, what is the length of side DF?

Introduction / Context:
This problem uses the idea of similar triangles created by a line parallel to one side of a triangle. It focuses on how a point dividing one side in a given ratio leads to proportional segments on the other sides when a parallel line is drawn.

Given Data / Assumptions:
- Triangle DEF is given.
- Point G lies on side DE and point H lies on side DF.
- Segment GH is parallel to side EF.
- G divides DE in the ratio DG : GE = 3 : 2, so DE is split into parts 3k and 2k for some k.
- HF, the segment from H to F on DF, is 8 cm.
- We must find the length of side DF.

Concept / Approach:
Because GH is parallel to EF, triangle DGH is similar to triangle DEF. In similar triangles, corresponding side lengths are proportional. The ratio DG / DE gives the scale factor between the smaller triangle DGH and the original triangle DEF. Once we know what fraction of DF corresponds to DH, we can relate HF to DF, since HF is the remaining part of DF after removing DH.

Step-by-Step Solution:
Step 1: From DG : GE = 3 : 2, write DG = 3k and GE = 2k. Step 2: Then DE = DG + GE = 3k + 2k = 5k. Step 3: Because GH ∥ EF, triangles DGH and DEF are similar, and the scale factor is DG / DE = 3k / 5k = 3 / 5. Step 4: Thus, corresponding sides satisfy DH / DF = 3 / 5, since DH in the smaller triangle corresponds to DF in the larger triangle. Step 5: Therefore DH = (3 / 5) * DF. Step 6: The remaining segment HF on DF is DF − DH = DF − (3 / 5) * DF = (2 / 5) * DF. Step 7: We are given HF = 8 cm, so (2 / 5) * DF = 8. Step 8: Solve for DF: DF = 8 * (5 / 2) = 20 cm.

Verification / Alternative Check:
If DF = 20 cm, then DH = (3 / 5) * 20 = 12 cm and HF = 20 − 12 = 8 cm, which matches the given length of HF. The internal ratios along the sides are consistent with the similarity factor 3 / 5, so the value DF = 20 cm is verified.

Why Other Options Are Wrong:
Option a, 12 cm, would give HF = (2 / 5) * 12 = 4.8 cm, not 8 cm.
Option c, 14 cm, would give HF = (2 / 5) * 14 = 5.6 cm, again not 8 cm.
Option d, 16 cm, would give HF = (2 / 5) * 16 = 6.4 cm, which also does not match the given value.

Common Pitfalls:
A common error is to use the ratio 3 : 2 incorrectly, for example using GE / DE instead of DG / DE, or to assume that HF corresponds directly to GH rather than to part of DF. Another pitfall is forgetting that the part of DF in the smaller triangle is DH, not HF. Careful labeling of the diagram and clear use of corresponding sides in similar triangles helps to avoid these mistakes.

Final Answer:
The length of DF is 20 cm.