Necessary and Sufficient Conditions for Fitting Regular Pentagon in The Right Triangle

Abstract

In this paper, we present necessary and sufficient conditions for fitting a regular pentagon which each side has the length a into the target right triangle  ABC with the right angle \angle ABC and the shorter leg, \overline{BC} , has the length c. We get the result as follow

(i) If \angle ACB = \alpha \leq 54^{\circ  then the regular pentagon can be fitted in the target triangle  if and only if  
a \leq \frac{c \tan \alpha}{2 \cos 36^{\circ} \sin \alpha + \cos 18^{\circ} \sec \alpha}.

(ii) if \angle ACB = \alpha > 54^{\circ}   then the regular pentagon can be fitted in the target triangle  if and only if
a \leq \frac{c \tan \alpha}{\cos 18^{\circ} \tan \alpha + \sin 36^{\circ} \tan \alpha + \cos 36^{circ}} .