# 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}} .

## Article Details

How to Cite
Anakkamatee, W., & Sueduang, W. (2019). Necessary and Sufficient Conditions for Fitting Regular Pentagon in The Right Triangle. Mathematical Journal by The Mathematical Association of Thailand Under The Patronage of His Majesty The King, 63(696), 22–34. Retrieved from https://ph02.tci-thaijo.org/index.php/MJMATh/article/view/193462
Section
Research Article

## References

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Chanakantha Meta. (2017). Necessary and sufficient conditions for fitting regular pentagon in the right triangle. (Master’s thesis, Naresuan University).
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