In triangle ABC, a line from vertex A meets side BC at point D. If BC = 9 cm and DC = 3 cm, what is the ratio of the areas of triangles ABD and ADC respectively?

Introduction / Context:
This geometry problem tests your understanding of how the areas of triangles that share the same altitude are related to the lengths of their bases. It appears frequently in aptitude exams because it can be solved quickly using proportional reasoning without any complicated formulas, provided you remember the basic area relation for triangles.

Given Data / Assumptions:
• Triangle ABC has a side BC of length 9 cm. • Point D lies on BC such that DC = 3 cm. • A line from vertex A meets BC at D, forming triangles ABD and ADC. • We need the ratio of areas of triangles ABD and ADC.

Concept / Approach:
The area of a triangle is given by (1 / 2) * base * height. If two triangles share the same altitude (height) from a common vertex to a common base line, then their areas are directly proportional to their bases. In this problem, both triangle ABD and triangle ADC have the same altitude from vertex A to line BC. Therefore, the ratio of their areas equals the ratio of their bases on BC, which are BD and DC. So we first find BD by subtracting DC from BC, then compare BD and DC directly.

Step-by-Step Solution:
1. Given BC = 9 cm and DC = 3 cm. 2. Since D lies on BC, we have BD + DC = BC. 3. Compute BD = BC - DC = 9 - 3 = 6 cm. 4. Area of triangle ABD is (1 / 2) * BD * height from A to BC. 5. Area of triangle ADC is (1 / 2) * DC * the same height from A to BC. 6. Let the common height from A to BC be h. Then area(ABD) = (1 / 2) * 6 * h and area(ADC) = (1 / 2) * 3 * h. 7. The ratio of the areas is area(ABD) : area(ADC) = (6 * h) : (3 * h). 8. Cancel h and common factor 3 to get 6 : 3 = 2 : 1.

Verification / Alternative check:
You can verify with simple numbers. Assume the height from A to BC is, for example, 4 cm. Then area(ABD) = (1 / 2) * 6 * 4 = 12 square centimetres and area(ADC) = (1 / 2) * 3 * 4 = 6 square centimetres. The ratio 12 : 6 simplifies to 2 : 1, which matches our algebraic result. This confirms that the reasoning using proportional bases is correct and does not depend on the actual height value.

Why Other Options Are Wrong:
• 1 : 1: This would mean BD and DC are equal, but BD = 6 cm and DC = 3 cm, so they are not. • 3 : 1: This would require BD to be three times DC, that is 9 cm versus 3 cm, which is not the case. • 4 : 1: This would correspond to BD = 12 cm and DC = 3 cm or a similar proportion, which does not match BC = 9 cm. • 5 : 1: Again, this does not reflect the actual ratio of 6 cm to 3 cm.

Common Pitfalls:
Some students mistakenly assume that the ratio of areas is related to the lengths of AD or to some angle measures, which is not needed. Others might incorrectly think that additional data is required. Remember that when two triangles share the same altitude to a common base line, the base lengths alone determine the area ratio. Drawing a rough figure often helps to visualise this clearly.

Final Answer:
The ratio of the areas of triangles ABD and ADC is 2 : 1.