In two triangles, the ratio of their areas is 4 : 3.\nThe ratio of their corresponding heights is 3 : 4. What is the ratio of their corresponding bases?

Introduction / Context:
Ratios of triangle areas, heights, and bases are tightly linked by the standard area formula. When two triangles are compared using corresponding elements, proportional reasoning quickly yields the missing ratio without requiring actual numeric side lengths.

Given Data / Assumptions:

• Area ratio: A1 : A2 = 4 : 3.
• Height ratio (corresponding heights): h1 : h2 = 3 : 4.
• Bases compared are the bases corresponding to those heights.

Concept / Approach:
For any triangle, area A = (1/2) * base * height. For two triangles using corresponding base and height, A1/A2 = (b1 * h1) / (b2 * h2). Therefore b1/b2 = (A1/A2) / (h1/h2).

Step-by-Step Solution:

Verification / Alternative check:
Pick convenient numbers: let h1 = 3, h2 = 4. Choose bases b1 and b2 so that areas match the 4:3 ratio. If b1/b2 = 16/9, then A1 : A2 = (1/2)*b1*3 : (1/2)*b2*4 = (3b1) : (4b2) = 3*(16k) : 4*(9k) = 48k : 36k = 4 : 3, confirming consistency.

Why Other Options Are Wrong:

• 13:9: Does not satisfy (A1/A2)/(h1/h2) relationship.
• 15:9: Simplifies to 5:3, still inconsistent with required 16:9.
• 14:9: No proportional derivation yields this pair.

Common Pitfalls:

• Multiplying the given ratios instead of dividing, which would produce 12:12 erroneously.
• Inverting the height ratio, leading to 9:16.

Final Answer:
16:9