Sides of two similar triangles are in the ratio $4:9$. Areas of these triangles are in the ratio

State which pairs of triangles in the figure are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form.

In the figure,$E$ is a point on the side $CB$ produced of an isosceles triangle $ABC$ with $AB = AC$. If $AD \perp BC$ and $EF \perp AC$,prove that $\Delta ABD \sim \Delta ECF$.

$ABC$ is an isosceles triangle with $AC = BC$. If $AB^2 = 2 AC^2$,prove that $ABC$ is a right triangle.

State which pairs of triangles in the figure are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form.

Nazima is fly fishing in a stream. The tip of her fishing rod is $1.8 \, m$ above the surface of the water and the fly at the end of the string rests on the water $2.4 \, m$ away from a point directly under the tip of the rod. Assuming that her string (from the tip of her rod to the fly) is taut,how much string does she have out? If she pulls in the string at the rate of $5 \, cm$ per second,what will be the horizontal distance of the fly from her after $12 \, seconds$?