In triangle ABC, points D and E lie on sides AB and BC respectively such that segment DE is parallel to side AC. If AB = 5 centimetres and AD = 3 centimetres, then what is the ratio BE : EC?

Introduction / Context:
This question applies the basic proportionality theorem (Thales theorem) in a triangle. When a line is drawn parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally. Using this result, we can find the ratio in which a side is divided when we know the division on another side.

Given Data / Assumptions:

• Triangle ABC is given.
• D lies on side AB and E lies on side BC.
• Segment DE is parallel to side AC.
• AB = 5 cm and AD = 3 cm.
• We need to find the ratio BE : EC.
• Standard Euclidean triangle geometry is assumed.

Concept / Approach:
By the basic proportionality theorem, if a line is drawn parallel to one side of a triangle and intersects the other two sides, then it divides those sides in the same ratio. Here DE is parallel to AC, so the segments on AB and BC are proportional:
AD / DB = BE / EC.
We know AB and AD, so we can find DB and then use the proportion to get BE : EC as a simple ratio.

Step-by-Step Solution:
Step 1: Compute DB from the total length AB and AD. Step 2: AB = 5 cm and AD = 3 cm, so DB = AB − AD = 5 − 3 = 2 cm. Step 3: By the basic proportionality theorem, AD / DB = BE / EC. Step 4: Substitute the known values: 3 / 2 = BE / EC. Step 5: Therefore the ratio BE : EC = 3 : 2.

Verification / Alternative check:
We can think in terms of a scale factor. If AB is partitioned in the ratio AD : DB = 3 : 2, then BC must be partitioned by point E in the same ratio for DE to be parallel to AC. Any specific choice of BC length, say BC = 5k, would give BE = 3k and EC = 2k, which maintains the 3 : 2 ratio and confirms that BE : EC must be 3 : 2.

Why Other Options Are Wrong:
2 : 3 and 3 : 5: These invert or distort the required proportion and would not maintain parallelism between DE and AC.
5 : 3 and 4 : 1: These ratios do not correspond to AD / DB = 3 / 2 and break the similarity between the smaller and larger triangles created by the parallel line.

Common Pitfalls:
A frequent mistake is to use AD / AB instead of AD / DB in the proportionality relation. Another pitfall is incorrectly writing the ratio as EC : BE rather than BE : EC, which reverses the result. Careful attention to which segments correspond and writing the proportionality in the correct order helps avoid these errors.

Final Answer:
The required ratio is BE : EC = 3 : 2.