State the two properties which are necessary for given two triangles to be similar.

In a ΔABC, AD is a median and AL ⊥ BC.

Prove that:

(a) AC² = AD² + BC • DL +

(b) AB² = AD² - BC • DL +

(c) AC² + AB² = 2AD² +

Naman is doing fly-fishing in a stream. The tip of his 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 3.6 m away from him and 2.4 m from the point directly under the tip of the rod. Assuming that the string (from the tip of his rod to the fly) is taut, how much string does he have out (see the adjoining figure)? If he pulls in the string at the rate of 5 cm per second, what will be the horizontal distance of the fly from him after 12 seconds?

Ans. 2.8 m (approx.)

State the basic proportionality theorem.

State the converse of Thales’ theorem.