In triangle DEF, points G and H lie on sides DE and DF respectively, and segment GH is parallel to side EF. Point G divides side DE in the ratio 1 : 3 and segment HF is 7.2 centimetres long. Find the length of side DF in centimetres.

Introduction / Context:
This question uses the concept of similar triangles formed when a line segment inside a triangle is drawn parallel to one of its sides. It combines ratio, similarity, and length calculations, which are common themes in geometry based aptitude problems.

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 1 : 3, meaning DG : GE = 1 : 3.
• Length of HF = 7.2 cm.
• We must find the total length of DF.

Concept / Approach:
Since GH is parallel to EF, triangles DGH and DEF are similar. This implies that corresponding sides of these triangles are in the same ratio. If DG : DE = 1 : 4 (because 1 + 3 = 4), then DH : DF also equals 1 : 4. Using this, we can express DF in terms of DH, and then relate DH and HF because DF = DH + HF.

Step-by-Step Solution:
Step 1: Ratio DG : GE = 1 : 3, hence DG : DE = 1 : (1 + 3) = 1 : 4. Step 2: Because triangles DGH and DEF are similar, the ratio DH : DF is the same as DG : DE. Step 3: Therefore, DH : DF = 1 : 4, which implies DH = DF / 4. Step 4: The full side DF is made of DH and HF: DF = DH + HF. Step 5: Substitute DH = DF / 4 into this: DF = DF / 4 + HF. Step 6: Rearrange: DF - DF / 4 = HF, so (3 / 4) * DF = HF. Step 7: HF is given as 7.2 cm, hence (3 / 4) * DF = 7.2. Step 8: Solve for DF: DF = 7.2 * (4 / 3) = 9.6 cm.

Verification / Alternative check:
Check DH: DH = DF / 4 = 9.6 / 4 = 2.4 cm. Then DF = DH + HF = 2.4 + 7.2 = 9.6 cm, which is consistent. Also, the ratio DH : DF = 2.4 : 9.6 = 1 : 4, matching the ratio obtained from the division of DE. This confirms the correctness of the answer.

Why Other Options Are Wrong:
2.4 cm: This is only DH, not the full side DF.
3.6 cm and 4.8 cm: These values are too small to accommodate HF = 7.2 cm when DF must equal DH + HF.
12 cm: This makes DH = 3 cm and HF = 7.2 cm, which does not satisfy the 1 : 4 ratio between DH and DF.

Common Pitfalls:
A frequent mistake is to assume the ratio 1 : 3 directly compares DE and DG rather than DG and GE. Another error is forgetting that the full side DF includes both DH and HF. Some learners also misapply the concept of similar triangles and invert the ratios. Always carefully define which segments correspond and verify proportional relationships.

Final Answer:
The length of side DF is 9.6 cm.