Question : If the ratio of the altitudes of two triangles is 3 : 4 and the ratio of their corresponding areas is 4 : 3, then the ratio of their corresponding lengths of bases is:
Option 1: 1 : 1
Option 2: 16 : 9
Option 3: 1 : 2
Option 4: 2 : 1
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Correct Answer: 16 : 9
Solution :
Given: the ratio of the altitudes of two triangles is 3 : 4 and the ratio of their corresponding areas is 4 : 3.
Let the altitudes of two triangles be 3$x$ and 4$x$
and the areas of the triangles be 4$y$ and 3$y$, respectively.
Also, their corresponding lengths of bases be $a$ and $b$, respectively.
According to the question,
$\frac{\frac{1}{2}×a×3x}{\frac{1}{2}×b×4x}=\frac{4y}{3y}$
⇒ $\frac{a}{b}=\frac{4}{3}×\frac{4}{3}$
⇒ $\frac{a}{b}=\frac{16}{9}$
Therefore, the ratio of their corresponding lengths of bases is 16 : 9.
Hence, the correct answer is 16 : 9.
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