The areas of two equilateral triangles are in the ratio 25 : 36. Their altitudes will be in the ratio :

Solution (By Examveda Team)

Let the length of sides of the two triangles be a₁ and a₂ respectively and their altitudes be h₁ and h₂ respectively.
Then,
$$\eqalign{ & \Leftrightarrow \frac{{\frac{{\sqrt 3 }}{4}a_1^2}}{{\frac{{\sqrt 3 }}{4}a_2^2}} = \frac{{25}}{{36}} \cr & \Rightarrow {\left( {\frac{{{a_1}}}{{{a_2}}}} \right)^2} = {\left( {\frac{5}{6}} \right)^2} \cr & \Rightarrow \frac{{{a_1}}}{{{a_2}}} = \frac{5}{6} \cr} $$
And
$$\eqalign{ & \Leftrightarrow \frac{{\frac{1}{2} \times {a_1} \times {h_1}}}{{\frac{1}{2} \times {a_2} \times {h_2}}} = \frac{{25}}{{36}} \cr & \Rightarrow \frac{5}{6} \times \frac{{{h_1}}}{{{h_2}}} = \frac{{25}}{{36}} \cr & \Rightarrow \frac{{{h_1}}}{{{h_2}}} = \frac{{25}}{{36}} \times \frac{6}{5} \cr & \Rightarrow \frac{{{h_1}}}{{{h_2}}} = \frac{5}{6} \cr} $$