In triangle ABC, points D and E lie on sides AB and AC respectively such that segment DE is parallel to side BC. If AD = 8 cm, DB = 4 cm and AE = 12 cm, then what is the total length, in centimetres, of side AC?

Introduction / Context:
This problem uses the concept of similar triangles that arise when a line is drawn parallel to one side of a triangle and intersects the other two sides. Such questions are very standard in geometry because they involve proportional reasoning and help reinforce understanding of basic similarity rules such as the Basic Proportionality Theorem (Thales theorem).

Given Data / Assumptions:

• In triangle ABC, D lies on side AB and E lies on side AC.

• Line segment DE is parallel to side BC.

• AD = 8 cm and DB = 4 cm, so AB = AD + DB = 8 + 4 cm.

• AE = 12 cm.

• All lengths are in centimetres.

Concept / Approach:
When a line is drawn parallel to one side of a triangle and cuts the other two sides, it forms a smaller triangle inside that is similar to the original triangle. In this case, triangle ADE is similar to triangle ABC. For similar triangles, the ratio of corresponding sides is equal. Thus AD / AB = AE / AC = DE / BC. We know AD, DB and AE, so we can first find AB and then use the side ratio to determine AC.

Step-by-Step Solution:
AD = 8 cm, DB = 4 cm. Therefore, AB = AD + DB = 8 + 4 = 12 cm. Because DE is parallel to BC, triangle ADE is similar to triangle ABC. So, AD / AB = AE / AC. Substitute the known values: 8 / 12 = 12 / AC. Simplify 8 / 12 = 2 / 3. Thus, 2 / 3 = 12 / AC. Cross multiply: 2 * AC = 3 * 12. 2 * AC = 36, so AC = 36 / 2 = 18 cm.

Verification / Alternative check:
We can check consistency by observing the scale factor between the smaller and larger triangles. From AD / AB = 8 / 12 = 2 / 3, the scale factor from triangle ABC to triangle ADE for corresponding sides is 2 / 3. That means every side of triangle ADE is 2 / 3 of the corresponding side of triangle ABC. So AE should equal (2 / 3) * AC. Reversing this, AC = (3 / 2) * AE = (3 / 2) * 12 = 18 cm, which matches our earlier result.

Why Other Options Are Wrong:
The value 6 cm is far too small and would not be consistent with AE = 12 cm. The value 9 cm would make AE longer than AC, which is impossible. The value 15 cm does not satisfy the similarity ratio AD / AB = AE / AC. Only 18 cm maintains the correct proportion between the small triangle ADE and the larger triangle ABC.

Common Pitfalls:
A frequent mistake is to add or subtract lengths incorrectly or to think in terms of AD : DB instead of AD : AB when setting up the similarity ratio. Some students also invert the ratio accidentally and solve 12 / AC = 3 / 2 instead of 2 / 3, which yields a wrong answer. Always confirm which sides are corresponding and use the entire side AB, not just the segment DB, when relating AD to AB.

Final Answer:
The total length of side AC in triangle ABC is 18 cm.