Math, asked by ngbhavyashree, 1 year ago

The circumference of a circle is divided into 6 parts such that The lengths are in AP if the smallest and largest arcs are in the ratio 1:5, then find the angle substended by the smallest arc.

Answers

Answered by Anonymous
13

\sf\red{\underline{\underline{Answer:}}}

\sf{The \ angle \ substended \ by \ smallest \ arc}

\sf{is \ 20^\circ.}

\sf{\longmapsto{The \ circumference \ of \ a \ circle \ is}}

\sf{divided \ into \ 6 \ parts \ such \ that \ the \ lengths}

\sf{are \ in \ AP.}

\sf{\longmapsto{The \ smallest \ arc \ and \ the \ largest \ arc}}

\sf{are \ in \ the \ ratio \ of \ 1:5}

\sf\pink{To \ find:}

\sf{The \ angle \ substended \ by \ the \ smallest \ arc.}

\sf\green{\underline{\underline{Solution:}}}

\sf{Let \ the \ lengths \ of \ 6 \ arcs \ be}

\sf{a-5d, \ a-3d, \ a-d, \ a+d, \ a+3d \ and \ a+5d}

\sf{We, \ know \ sum \ of \ 6 \ parts=Circumference}

\sf{\mapsto{(a-5d)+(a-3d)+(a-d)+(a+d)+(a+3d)+(a+5d)=2\pi \times \ r}}

\sf{\mapsto{\therefore{6a=2\pi\times \ r}}}

\sf{\mapsto{\therefore{a=\dfrac{\pi\times \ r}{3}...(1)}}}

\sf{According \ to \ the \ given \ condition. }

\sf{\dfrac{a-5d}{a+5d}=\dfrac{1}{5}}

\sf{\therefore{5(a-5d)=a+5d}}

\sf{But \ a=\dfrac{\pi\times \ r}{3}}

\sf{\therefore{\dfrac{5\pi\times \ r}{3}-25d=\dfrac{pi\times \ r}{3}+5d}}

\sf{\therefore{30d=\dfrac{5\pi\times \ r}{3}-\dfrac{pi\times \ r}{3}}}

\sf{\therefore{30d=\dfrac{4\pi\times \ r}{3}}}

\sf{\therefore{d=\dfrac{2\pi\times \ r}{45}}}

\sf{The \ smallest \ arc=a-5d}

\sf{=\dfrac{\pi\times \ r}{3}-5(\dfrac{2\pi\times \ r}{45})}

\sf{=\dfrac{\pi\times \ r}{3}-\dfrac{2\pi\times \ r}{9}}

\sf{=\dfrac{3\pi\times \ r-2\pi\times \ r}{9}}

\sf{=\dfrac{\pi\times \ r}{9}}

\sf{We \ know,}

\sf{Length \ of \ arc=\theta^{c}\times \ r}

\sf{=\dfrac{\pi}{9}\times \ r}

\sf{On \ comparing \ we \ get,}

\sf{\theta^{c}=\dfrac{\pi}{9}}

\sf{\theta^{c}=(\theta\times\dfrac{180}{\pi})^\circ}

\sf{=\dfrac{\pi}{9}\times\dfrac{180}{\pi}}

\sf{=\dfrac{180}{9}}

\sf{=20^\circ}

\sf\purple{\tt{\therefore{The \ angle \ substended \ by \ smallest \ arc}}}

\sf\purple{\tt{is \ 20^\circ.}}

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