A birthday conical cap is cut by a plane parallel to its base and the upper part is used as a new cap for a toy. The curved surface area of this new cap is \[\dfrac{1}{9}\] of the curved surface area of the whole cone. Find the ratio of the line segments into which the cone’s altitude is divided by the plane.

Hint: To solve this question we will, first of all, assume the radius, height and slant height of the small cone and the bigger cone and then try to calculate the radius of both radius and height and slant height of the both. Then using the curved surface area of the cone, \[LSA=\pi rl,\] where the radius is ‘r’ and slant height is ‘l’. Finally, using AA symmetry, we will get the required ratio.

Complete step-by-step answer:
Let us assume the radius of the conical cap be R and the height of the conical cap be H and the slant height of the conical cap be L. Again let us assume the radius of the new cone be r, the height of the new cone (cap) be h and the slant height of the new cone be l. The figure for the above situation is given as

The formula of the curved surface area of the cone is given by \[\pi rl\] where r is the radius of the base and l is the slant height of the cone. According to the given condition in the question, we have,
\[\text{Curved surface area of new cap}=\dfrac{1}{9}\text{ curved surface area of whole cone}\]
Using the above stated formula we have,
\[\text{Curved surface area of new cap}\left( \text{CSA} \right)=\pi rl\]
\[\text{Curved surface area of whole cone}=\pi RL\]
Applying this in the above, we get,
\[\Rightarrow \pi rl=\dfrac{1}{9}\pi RL\]
Cancelling \[\pi \] on both sides,
\[\Rightarrow rl=\dfrac{RL}{9}\]
\[\Rightarrow \dfrac{r}{R}\times \dfrac{l}{L}=\dfrac{1}{9}......\left( i \right)\]
From the figure,

Now from the figure, in triangle ADE and triangle ABC,
\[\angle A=\angle A\left[ \text{Common} \right]\]
\[\angle ADE=\angle ABC\left[ \text{Corresponding angles as DE }\!\!|\!\!\text{ }\!\!|\!\!\text{ BC} \right]\]
By using AA symmetry (Angle – Angle Symmetry)
\[\Delta ADE\sim \Delta ABC\]
Also,
\[\dfrac{r}{R}=\dfrac{h}{H}=\dfrac{l}{L}\left[ \text{By CPCT} \right]\]
Substituting the value of \[\dfrac{r}{R}\] as \[\dfrac{h}{H}\] in equation (i), we get,
\[\Rightarrow \dfrac{r}{R}\times \dfrac{h}{H}=\dfrac{1}{9}\]
\[\Rightarrow \dfrac{h}{H}\times \dfrac{h}{H}=\dfrac{1}{9}\]
\[\Rightarrow \dfrac{{{h}^{2}}}{{{H}^{2}}}=\dfrac{1}{9}\]
Taking the square root, we have,
\[\Rightarrow \dfrac{h}{H}=\dfrac{1}{3}\]
\[\Rightarrow \dfrac{H}{h}=\dfrac{3}{1}\]
Now, subtracting 1 both the sides of the above equation, we have,
\[\Rightarrow \dfrac{H}{h}-1=\dfrac{3-1}{1}\]
\[\Rightarrow \dfrac{H-h}{h}=\dfrac{2}{1}\]
\[\Rightarrow H-h:H=2:1\]
Hence, the ratio of the line segments in which the cone’s altitude is divided by the plane is 2:1.

Note: AA symmetry is stated as “When two angles of two different triangles are equal, then according to AA symmetry, both the triangles are similar and CPCT is stated as a correspondent part of the congruent triangle.