Question

plzz solve it with explanation!!!​

Answer 1

Given :-

• Ratio between the exterior angle and interior angle of a regular polygon = 2 : 3

To Find :-

• The number of sides and name of the polygon

Solution :-

• Now , each interior angle of a regular polygon is given by ;

$\\ \star \: {\boxed{\purple{\sf{ Each \: interior \: angle_{(regular \: polygon)} = \dfrac{ {180}^{ \circ} (n - 2)}{n} }}}}$

Where ,

• n is number of sides of the polygon

• Now , Each Exterior angle of a regular polygon is given by ;

$\\ \star \: {\boxed{\sf{\purple{Each \: exterior \: angle_{(regular \: polygon)} \: = \: \dfrac{ {360}^{ \circ} }{n} }}}}$

Where ,

n is number of sides of the polygon

According to the question :-

$\\ : \implies \sf \: \dfrac{ {360}^{ \circ} }{n} : \dfrac{ {180}^{ \circ}(n - 2) }{n} = 2 : 3$

$\\ : \implies \sf \: \dfrac{360}{n} \times \frac{n}{180(n - 2)} = \dfrac{2}{3}$

$\\ : \implies \sf \: \dfrac{ 360 }{180(n - 2)} = \dfrac{2}{3}$

$\\ : \implies \sf \: {180}^{ \circ}(n - 2) \times 2 = 3 \times {360}^{ \circ}$

$\\ : \implies \sf \: 180 n - 360 \times 2= 1080$

$\\ : \implies \sf \: 180n - 360 = \dfrac{1080}{2}$

$\\ : \implies \sf \: 180n - 360 = 540$

$\\ : \implies \sf \: 180n = 540 + 360$

$\\ : \implies \sf \: 180n = 900$

$\\ : \implies{\underline{\boxed {\pink{\mathfrak{n = 5}}}}} \: \bigstar$

Hence ,

• The number of sides of the given polygon are 5 . So , the name of the given regular polygon is pentagon.

Answer 2

${\huge{\underline{\boxed{\red{\mathcal{Answer}}}}}}$

Given:

• It is a regular polygon.
• Ratio of exterior angle to interior angle is 2:3.

To Find:

• Number of sides of the polygon.
• Name of the polygon.

Solution:

Exterior Angle = $\dfrac{360}{n}$

Interior Angle = $\dfrac{180(n-2)}{n}$

Where n = no of sides.

So,

$\implies \dfrac {\dfrac{360}{n} } { \dfrac{180(n-2)}{n}} = \dfrac{2}{3}$

$\implies \dfrac {360} {180(n-2)} = \dfrac{2}{3}$

$\implies \dfrac{2}{n-2} = \dfrac{2}{3} \\ \implies 3 = n - 2 \\ \implies n = 5.$

The polygon has 5 sides and is a pentagon.

Formulae Used:

• Exterior Angle = $\dfrac{360}{n}$
• Interior Angle = $\dfrac{180(n-2)}{n}$

HOPE IT HELPS!!!!