In ΔABC, AB = AC and BC = 6 cm. D is a point on the side AC such that AD = 5 cm and CD = 4 cm. Show that ΔBCD ~ ΔACB and hence find BD.

A girl is in the beach with her father. She spots a swimmer drowning. She shouts to her father who is 50 m due west of her. Her father is 10 m nearer to a boat than the girl. If her father uses the boat to reach the swimmer, he has to travel a distance 126 m from that boat. At the same time, the girl spots a man riding a water craft who is 98 m away from the boat. The man on the water craft is due east of the swimmer. How far must the man travel to rescue the swimmer? (Hint: see figure).

P and Q are points on sides AB and AC respectively, of ΔABC. If AP = 3 cm, PB = 6 cm, AQ = 5 cm and QC = 10 cm, show that BC = 3 PQ.

The points D and E are on the sides AB and AC of ΔABC respectively, such that DE || BC. If AB = 3 AD and the area of Δ ABC is 72 cm², then find the area of the quadrilateral DBCE.

The lengths of three sides of a triangle ABC are 6 cm, 4 cm and 9 cm. ΔPQR ~ ΔABC. One of the lengths of sides of ΔPQR is 35cm. What is the greatest perimeter possible for ΔPQR?