Question

the difference between an exterior angle of a regilar polygon of n sides and an exterior angle of a regilar polygon (n+1) sides is equal to 18 °. Find the number of sides of each polygon. ​

Answer 1

Answer:

Sum of exterior angles of a polygon=360

∘

An n sided polygon has n equal exterior angles.

n

360

∘

−

n+1

360

∘

=4

∘

⇒360

∘

[

n

1

−

n+1

1

]=4

∘

⇒20

∘

[

n(n+1)

n+1−n

]=1

⇒n

2

+n−20

⇒n

2

+5n−4n−20=0

⇒n(n+5)−4(n+5)=0

⇒n=4,-5

∴n=4

Answer 2

Step-by-step explanation:

$\sf { \large \bold\green{ \mathbb{GIVEN}}}$

$\sf \: Difference \: \: in \: \: angles = 18 \degree$

$\sf \: We \: know \: that \: the \: sum \: of \\ \sf \: exterior \: angles \: of \: polygon = 360 \degree$

$\rightarrow \sf \: According \: to \: Question \leftarrow$

$\sf \: difference \: in \: angle \: = 18 \degree \\ \alpha - \beta = 18 \degree \\ \\ \sf \: \: \implies\frac{360}{n} - \frac{360}{n + 1} = 18 \\ \\ \sf \: \implies360 \bigg( \frac{1}{n} - \frac{1}{n + 1} \bigg) = 18 \\ \\ \sf \implies \: \frac{n + 1 - n}{n(n + 1)} = \frac{18}{360} \\ \\ \sf \implies \: \frac{1}{n(n + 1)} = \frac{1}{20} \: \: \: \\ \\ \sf \: \implies \: n(n + 1) = 20 \\ \\ \sf \implies \: {n}^{2} + n - 20 = 0 \\ \\ \sf \implies \: {n}^{2} + 5n - 4n - 20 = 0 \\ \\ \sf \implies n(n + 5) - 4(n + 5) = 0 \\ \\ \sf \implies (n + 5)(n - 4) = 0 \\ \\ \sf \implies n = 4 \: \: or \: - 5$

$\sf \implies since \: number \: of \: sides \: can \: not \: be \: in \: neg.$

$\sf \implies \: hence \: \green{ \boxed { \sf \: n = 4}}$