Question

Q25. Two regular polygons are
such that the ration between the number of sides is 1:2 and
interior angles is 3:4 Find the number of sides of each polygon.​

Answer 1

AnswEr :

• Ratio of Number of Sides of Polygons is 1:2.
• Ratio of Each Interior Angle of Polygons is 3:4.
• Find the Number of Sides of Each Polygons.

• Let the Sides be a and, 2a of Polygons.

$\small\boxed{ \bf{Each \: Interior \: Angle = \dfrac{(2 \times \:Side - 4) \times 90 }{Side} }}$

• Now Let's Head to the Question :

$\longrightarrow \sf{ \dfrac{Interior \: Angle \: Of \: Polygon_1}{Interior \: Angle \: Of \: Polygon_2} = \dfrac{3}{4} }$

$\longrightarrow \sf{ \dfrac{ \frac{(2 \times Side_1 - 4) \times \cancel{90}}{Side_1} }{\frac{(2 \times Side_2 - 4) \times \cancel{90}}{Side_2}} = \dfrac{3}{4} }$

$\longrightarrow \sf{ \dfrac{ \frac{(2 \times a - 4)}{\cancel{a}} }{\frac{(2 \times 2a - 4) }{2 \cancel{a}}} = \dfrac{3}{4} }$

$\longrightarrow \sf{ \dfrac{(2a - 4) \times 2}{(4a - 4)} = \dfrac{3}{4} }$

$\longrightarrow \sf{ \dfrac{(4a - 8)}{(4a - 4)} = \dfrac{3}{4} }$

$\longrightarrow \sf{4 \times (4a - 8) = 3 \times (4a - 4)}$

$\longrightarrow \sf{16a - 32 = 12a - 12}$

$\longrightarrow \sf{16a - 12a = 32 - 12}$

$\longrightarrow \sf{4a = 20}$

$\longrightarrow \sf{a = \cancel\dfrac{20}{4} }$

$\longrightarrow \large\boxed{ \sf{a = 5}}$

⋆ Sides of Polygon₁ = a = 5

⋆ Sides of Polygon₂ = 2a = ( 2 × 5 ) = 10

⠀

჻ Sides of Polygⁿ are 5 & 10 respectively.