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.
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Aay:bx
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Ba:b
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Cx:y
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Dbx:ay
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ENone of these
Answer
Correct Answer: ay:bx
Explanation
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:
a/b = (x * h1) / (y * h2) ⇒ h1/h2 = (a/b) * (y/x) = ay / (bx)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