Question

ABCDE is a regular pentagon with sides of length 6 cm. CD is also a side of a regular polygon with n sides. Given that \[\angle {\text{EDF = 9}}{{\text{0}}^{\text{0}}}\], find \[n\].

A. 18
B. 10
C. 20
D. 12

Answer 1

Hint: Whenever there is a question on calculating the sides pentagon, we can calculate by knowing the angles of the pentagon i.e. \[{108^0}\]. This will help us to calculate the remaining angles and thus sides.

Complete step-by-step answer:
ABCDE is a pentagon. Hence,
\[\angle {\text{EDC = 10}}{{\text{8}}^{\text{0}}}\] (Interior angle of a pentagon)
Also, \[\angle {\text{EDF = 9}}{{\text{0}}^{\text{0}}}\]
Now, \[\angle {\text{CDF = 36}}{{\text{0}}^{\text{0}}}{\text{ - }}\angle {\text{EDF - }}\angle {\text{EDC}}\]
\[\angle {\text{CDF = 36}}{{\text{0}}^{\text{0}}} - {90^0} - {108^0} = {162^0}\]
Now, Interior angle of a Polygon,
\[\dfrac{{{{180}^0}(n - 2)}}{n} = {162^0}\]
\[180n - 360 = 162n\]
\[18n = 360\]
\[n = 20\].
Thus, Polygon has 20 sides.

Note: In this type of problems, we need to know the interior angle of the pentagon which is always \[{108^0}\]which in turn helps us to find the remaining angles. In the above question the formula for interior angle of Polygon i.e. \[\dfrac{{{{180}^0}(n - 2)}}{n}\]is used to find the sides of a polygon. It is wise to remember this formula for this type of questions.