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

A. 25 : 36

B. 36 : 25

C. 5 : 6

D. $\sqrt{5}$ : $\sqrt{6}$

Solution:

Let the length of sides of the two triangles be a₁ and a₂ respectively and their altitudes be h₁ and h₂ respectively.
Then,

\begin{aligned} \Leftrightarrow \frac{\frac{\sqrt{3}}{4} a_{1}^{2}}{\frac{\sqrt{3}}{4} a_{2}^{2}} = \frac{25}{36} \\ \Rightarrow {(\frac{a_{1}}{a_{2}})}^{2} = {(\frac{5}{6})}^{2} \\ \Rightarrow \frac{a_{1}}{a_{2}} = \frac{5}{6} \end{aligned}

And

\begin{aligned} \Leftrightarrow \frac{\frac{1}{2} \times a_{1} \times h_{1}}{\frac{1}{2} \times a_{2} \times h_{2}} = \frac{25}{36} \\ \Rightarrow \frac{5}{6} \times \frac{h_{1}}{h_{2}} = \frac{25}{36} \\ \Rightarrow \frac{h_{1}}{h_{2}} = \frac{25}{36} \times \frac{6}{5} \\ \Rightarrow \frac{h_{1}}{h_{2}} = \frac{5}{6} \end{aligned}

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