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
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
