Given ABC with ∠B = 60° and ∠C = 30°, let P, Q, R be points on sides BA, AC, CB respectively such that 2  ABC  BPQR is an isosceles trapezium with PQ||BR and BP = QR. Find the maximum possible value of BPQR  where [S] denotes the area of any polygon S.

Step 1: BP = QR = CR = $x$ & let BC = 

Step 2: PQ = ( – x) – 2x cos60°

Step 3:  ABC  2. .

Step 4: 2 2 2

Step 5: 2  =

Step 6: BPQR   – $x$   – 2 $x$ x 3 $x$  2 – 3 $x$ 

Step 7:  

Step 8: 

Step 9: x $x$ in 0,

Step 10: = , 2

Step 11:  

Step 12: 2 – 3 in in 2,  

Step 13: $x$ x

Step 14: Let f(y) = , f (y) = 0 gives $y$ = 3 for minimum value.

Step 15: Minimum value of the expression

Step 16:  ABC   32  3

Step 17: BPQR  2.3 – 3

Step 18: Maximum value tends to infinity.