Relating areas, bases, and altitudes of two triangles The ratio of the bases of two triangles is x:y and the ratio of their areas is a:b. Find the ratio of their corresponding altitudes.

Introduction / Context:Area of a triangle equals (1/2) * base * altitude. Comparing two triangles lets us relate the ratios of areas, bases, and altitudes algebraically.

Given Data / Assumptions:

• Base ratio b1:b2 = x:y
• Area ratio A1:A2 = a:b
• Altitudes h1 and h2 correspond to b1 and b2 respectively.

Concept / Approach:Since A ∝ base * altitude, we have A1/A2 = (b1 * h1)/(b2 * h2). Rearranging gives the altitude ratio in terms of given ratios.

Step-by-Step Solution:

Verification / Alternative check:Plug numbers: if x=2, y=3, a=4, b=5, then h1/h2 = (4*3)/(5*2) = 12/10 = 6/5, consistent with formula.

Why Other Options Are Wrong:a:b ignores base effects; x:y ignores area differences; bx:ay is the inverse of the correct relation.

Common Pitfalls:Forgetting that the 1/2 factor cancels out or inverting the ratios during cross-multiplication leads to wrong answers.

Final Answer:ay:bx