In the figure shown, ABCDEF is a regular hexagon. What is the value of AB + AC + AD + AE + AF?

In the figure shown, ABCDEF is a regular hexagon. What is the value of AB + AC + AD + AE + AF?

A
AO
B
2AO
C
4AO
D
6AO

Solution:

AB+ AC+ AD+ AE+ AF

$= AB + (AB + BC) + (AB + BC + CD) + (AB + BC + CD + DE) + AF$

$= 4 AB + 3 BC + 2 CD + DE + AF$

$(∵ AB = - DE, BC = - EF, AF = CD)$

$= 3 (AB + BC + CD) = 3 AD = 6 AO$

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