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

Introduction / Context:
This question tests proportional reasoning using the triangle area formula. The area of a triangle is (1/2)*base*height. When comparing two triangles, the factor 1/2 cancels in the ratio, so the area ratio depends on the product of base and height. If the area ratio and height ratio are known, the base ratio can be isolated by dividing the area ratio by the height ratio (in multiplicative terms). This is a classic “ratio of products” problem and does not require any absolute measurements.

Given Data / Assumptions:

• Area1:Area2 = 4:3
• Height1:Height2 = 3:4
• Area proportionality: Area is proportional to base*height

Concept / Approach:
Use: Area1/Area2 = (b1*h1)/(b2*h2). Rearrange to get: b1/b2 = (Area1/Area2) * (h2/h1). Substitute the ratios and simplify.

Step-by-Step Solution:
Area1/Area2 = (b1*h1)/(b2*h2) So (4/3) = (b1/b2) * (h1/h2) Given h1/h2 = 3/4 (4/3) = (b1/b2) * (3/4) b1/b2 = (4/3) * (4/3) = 16/9 Base ratio = 16:9

Verification / Alternative check:
Pick b1:b2 = 16:9 and h1:h2 = 3:4. Then (b1*h1):(b2*h2) = (16*3):(9*4) = 48:36 = 4:3, matching the area ratio, so the result is consistent.

Why Other Options Are Wrong:
13:9, 14:9, 15:9 are close-looking but do not produce exact 4:3 when combined with 3:4 heights. 9:16 is the inverse of the correct ratio.

Common Pitfalls:
Forgetting that area is proportional to base*height, reversing height ratio, or incorrectly adding ratios instead of multiplying/dividing them.

Final Answer:
The ratio of their bases is 16:9.