Question

Pentagon ABCDE is such that all five diagonals AC, BD, CE, DA and EB lie entirely within it. If the area of each of the triangle ABC, BCD, CDE, DEA and EAB is equal to 1 cm², the area of the pentagon is :

(1) 5/√5/2 cm²

(2) 3 +√5/2 cm²

(3) 5+√5/2 cm²

(4) 5- √5/2 cm²

Answer 1

Correct option (3) 5+√5/2 cm²

Explanation:

ar (ABC) = 1/2 AB BC(sin(108°))

= 1/2 x² sin (108°)= 1

x² = 2/sin(108°)

Area of ABCDE = 10 (ar AOP) 

= 10(1/2 AP x tan 54 (AP))

= 5. AP² × tan 54

= 5.x²/4 tan 54 = 5/2 tan54/2sin108°

= 5+√5/2 cm²