Sides AB, BC and median AD of the triangle ABC are respectively proportional to sides PQ, QR and median PM of triangle PQR.

Prove that:

(i) Δ ABD ∼ Δ PQM

(ii) Δ ABC ∼ Δ PQR

Find the value of x, if ∠B = ∠E.

In Δ ABC and Δ DEF, AB = 3 x DF, BC = 3 x DE and AC = 3 x EF. Show that the given triangle are similar. Name the two similar triangle on a proper way.

In a trapezium ABCD, side AB is parallel to side DC; and the diagonals AC and BD intersect each other at point P. Prove that:

(i) ΔAPB is similar to ΔCPD.

(ii) PA x PD = PB x PC.