Problem: If a,b and d are the lengths of a side, a shortest diagonal and a longest diagonal, respectively, of a regular nonagon, then
Answer Choices:
A. d=a+b
B. d2=a2+b2
C. d2=a2+ab+b2
D. b=2a+d​
E. b2=ad
Solution:
In the adjoining figure, P1​P2​⋯P9​ is a regular nonagon; P1​P2​=a; P2​P4​=b;P1​P5​=d;Q and R lie on P1​P5​;P2​Q⊥P1​P5​;P4​R⊥P1​P5​. Since P2​P3​=P3​P4​ and the interior angles of a regular nonagon are each
(9180(9−2)​)∘=140∘
∠P3​P2​P4​=20∘. Hence ∠P1​P2​Q=30∘ and P1​Q=2a​. Similarly, P5​R=2a​. Thus d=a+b.
