If the areas of two similar triangles are in the ratio 9 : 64, then the ratio of their corresponding altitudes is:

Let the areas be A₁ and A₂, and the altitudes be h₁ and h₂ respectively.

Since the triangles are similar,

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides (or altitudes).

$$\begin{aligned}\Rightarrow \frac{A_1}{A_2} = \Big(\frac{h_1}{h_2}\Big)^2 \\[1em] \Rightarrow \frac{9}{64} = \Big(\frac{h_1}{h_2}\Big)^2 \\[1em] \Rightarrow \frac{h_1}{h_2} = \sqrt{\frac{9}{64}} \\[1em] \Rightarrow \frac{h_1}{h_2} = \frac{3}{8} \\[1em] \Rightarrow h_1 : h_2 = 3 : 8.\end{aligned}$$

Hence, option 1 is the correct option.