In triangle ABC, DE is drawn parallel to side BC. Given AB = 7.5 cm, BD = 6 cm on side AB, and DE = 2 cm, what is the length (in centimetres) of side BC?

Introduction / Context:
This geometry question uses the concept of similar triangles when a line segment is drawn parallel to one side of a triangle. When DE is parallel to BC in triangle ABC, it divides the other two sides proportionally, and the smaller triangle ADE is similar to the larger triangle ABC. The problem provides segment lengths along AB and the length of DE, and asks for the full length of side BC using these similarity relationships.

Given Data / Assumptions:
- Triangle ABC is given.- DE is drawn parallel to BC (DE || BC).- AB = 7.5 cm.- BD = 6 cm, so point D lies on AB between A and B.- DE = 2 cm.- We assume standard Euclidean geometry and basic theorems of similar triangles.

Concept / Approach:
Since DE is parallel to BC, triangle ADE is similar to triangle ABC by the Basic Proportionality Theorem or the AA similarity criterion. This implies that corresponding sides are in the same ratio. Specifically, AD / AB = DE / BC. We can find AD using AB and BD, then use the proportion AD / AB = DE / BC to determine BC. This is a straightforward application of properties of similar triangles and proportional segments created by a line parallel to one side of a triangle.

Step-by-Step Solution:
Step 1: Given AB = 7.5 cm and BD = 6 cm, compute AD.Step 2: Since D lies between A and B, AD = AB - BD = 7.5 - 6 = 1.5 cm.Step 3: Because DE || BC, triangles ADE and ABC are similar.Step 4: Use the similarity ratio: AD / AB = DE / BC.Step 5: Substitute known values: AD = 1.5, AB = 7.5, DE = 2, and BC is unknown.Step 6: So 1.5 / 7.5 = 2 / BC.Step 7: Simplify 1.5 / 7.5 = 1 / 5.Step 8: Therefore, 1 / 5 = 2 / BC.Step 9: Cross multiply: BC = 2 * 5 = 10 cm.

Verification / Alternative check:
We can verify the proportion by computing the scaling factor from the smaller triangle ADE to the larger triangle ABC. From AD = 1.5 to AB = 7.5, the scale factor is 7.5 / 1.5 = 5. Therefore every side in triangle ABC is 5 times the corresponding side in triangle ADE. Since DE is 2 cm, the corresponding side BC should be 2 * 5 = 10 cm. This agrees with our calculated answer and confirms the correctness of the logic used.

Why Other Options Are Wrong:
- 6: This is smaller than DE and does not respect the scaling factor obtained from the ratio AD / AB.- 8: This would correspond to a scale factor of 4 from DE to BC, which conflicts with the factor of 5 from AD to AB.- 10.5: This does not match any proportional relationship derived from the given segment lengths.- 12: This would imply an even larger scale factor inconsistent with the 1.5 to 7.5 ratio.

Common Pitfalls:
Students sometimes incorrectly assign segments to the wrong corresponding sides in similar triangles, or forget to subtract BD from AB to obtain AD. Another mistake is treating the ratio AB / AD instead of AD / AB, which leads to incorrect scaling factors. Careful identification of corresponding sides and correct usage of proportions is essential in solving such problems accurately.

Final Answer:
The length of side BC is 10 cm.