Prove that the area of a triangle with vertices is independent of t.

A bird is sitting on the top of a 80 m high tree. From a point on the ground, the angle of elevation of the bird is 45 �. The bird flies away horizontally in such a way that it remained at a constant height from the ground. After 2 seconds, the angle of elevation of the bird from the same point is 30 �. Find the speed of flying of the bird.

A thief runs with a uniform speed of 100 m/minute. After one minute a policeman runs after the thief to catch him. He goes with a speed of 100 m/minute in the first minute and increases his speed by 10 m/minute every succeeding minute. After how many minutes the policeman will catch the thief.

A game of chance consists of spinning an arrow on a circular board, divided into 8 equal parts, which comes to rest pointing at one of the numbers 1, 2, 3, …, 8 (Fig. 9), which are equally likely outcomes. What is the probability that the arrow will point at (i) an odd number (ii) a number greater than 3 (iii) a number less than 9.

An elastic belt is placed around the rim of a pulley of radius 5 cm. (Fig. 10) From one-point C on the belt, the elastic belt is pulled directly away from the center O of the pulley until it is at P, 10 cm from the point O. Find the length of belt that is still in contact with the pulley. Also find the shaded area.