Question and Answers Forum

All Questions      Topic List

Geometry Questions

Previous in All Question      Next in All Question      

Previous in Geometry      Next in Geometry      

Question Number 182474 by Acem last updated on 10/Dec/22

Find the number of sides of two regular polygons   that their sides has a ratio 5:4 and of 9° as a   difference between their angles.

$${Find}\:{the}\:{number}\:{of}\:{sides}\:{of}\:{two}\:{regular}\:{polygons} \\ $$$$\:{that}\:{their}\:{sides}\:{has}\:{a}\:{ratio}\:\mathrm{5}:\mathrm{4}\:{and}\:{of}\:\mathrm{9}°\:{as}\:{a} \\ $$$$\:{difference}\:{between}\:{their}\:{angles}. \\ $$

Answered by mr W last updated on 10/Dec/22

n−sided regular polygon:  angle θ_n =180−((360)/n)  say two regular polygons with x and  y sides respectively.  (x/y)=(5/4)  θ_x −θ_y =((360)/y)−((360)/x)=9  ⇒(1/y)−(1/x)=(1/(40))  ⇒(1/y)−(4/(5y))=(1/(40)) ⇒y=8 ⇒x=10  i.e. their number of sides is 10 and 8  respectively.

$${n}−{sided}\:{regular}\:{polygon}: \\ $$$${angle}\:\theta_{{n}} =\mathrm{180}−\frac{\mathrm{360}}{{n}} \\ $$$${say}\:{two}\:{regular}\:{polygons}\:{with}\:{x}\:{and} \\ $$$${y}\:{sides}\:{respectively}. \\ $$$$\frac{{x}}{{y}}=\frac{\mathrm{5}}{\mathrm{4}} \\ $$$$\theta_{{x}} −\theta_{{y}} =\frac{\mathrm{360}}{{y}}−\frac{\mathrm{360}}{{x}}=\mathrm{9} \\ $$$$\Rightarrow\frac{\mathrm{1}}{{y}}−\frac{\mathrm{1}}{{x}}=\frac{\mathrm{1}}{\mathrm{40}} \\ $$$$\Rightarrow\frac{\mathrm{1}}{{y}}−\frac{\mathrm{4}}{\mathrm{5}{y}}=\frac{\mathrm{1}}{\mathrm{40}}\:\Rightarrow{y}=\mathrm{8}\:\Rightarrow{x}=\mathrm{10} \\ $$$${i}.{e}.\:{their}\:{number}\:{of}\:{sides}\:{is}\:\mathrm{10}\:{and}\:\mathrm{8} \\ $$$${respectively}. \\ $$

Commented by Acem last updated on 10/Dec/22

Thanks Sir!

$${Thanks}\:{Sir}! \\ $$

Answered by som(math1967) last updated on 10/Dec/22

let sides are 5x,4x   ⇒((360)/(4x)) −((360)/(5x))=9  ⇒((90−72)/x)=9  ⇒x=((18)/9)=2  ∴ sides are 10,8

$${let}\:{sides}\:{are}\:\mathrm{5}{x},\mathrm{4}{x} \\ $$$$\:\Rightarrow\frac{\mathrm{360}}{\mathrm{4}{x}}\:−\frac{\mathrm{360}}{\mathrm{5}{x}}=\mathrm{9} \\ $$$$\Rightarrow\frac{\mathrm{90}−\mathrm{72}}{{x}}=\mathrm{9} \\ $$$$\Rightarrow{x}=\frac{\mathrm{18}}{\mathrm{9}}=\mathrm{2} \\ $$$$\therefore\:{sides}\:{are}\:\mathrm{10},\mathrm{8} \\ $$

Commented by Acem last updated on 10/Dec/22

Thanksss Sir!

$${Thanksss}\:{Sir}! \\ $$

Commented by som(math1967) last updated on 10/Dec/22

����

Commented by Acem last updated on 10/Dec/22

Coeur  (:

$${Coeur}\:\:\left(:\right. \\ $$

Answered by Acem last updated on 10/Dec/22

ϕ_2 −ϕ_1 = ((360)/n_2 ) − ((360 ((4/5)))/n_2 )    ;   ϕ: external angle   9= ((72)/n_2 ) ,      n_2 = 8, n_1 = 10

$$\varphi_{\mathrm{2}} −\varphi_{\mathrm{1}} =\:\frac{\mathrm{360}}{{n}_{\mathrm{2}} }\:−\:\frac{\mathrm{360}\:\left(\frac{\mathrm{4}}{\mathrm{5}}\right)}{{n}_{\mathrm{2}} }\:\:\:\:;\:\:\:\varphi:\:{external}\:{angle} \\ $$$$\:\mathrm{9}=\:\frac{\mathrm{72}}{{n}_{\mathrm{2}} }\:,\:\:\:\:\:\:{n}_{\mathrm{2}} =\:\mathrm{8},\:{n}_{\mathrm{1}} =\:\mathrm{10} \\ $$$$\: \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com