Math, asked by ajay88888, 8 months ago

the angles of a triangle are in AP and the greatest angle is double the least. find all the angles in degrees and radians​

Answers

Answered by Anonymous
81

\huge\bf\red{\underline{\underline{Given}}}\::

  • \sf\gray{The\: angles\: of\: a\: triangle\: are \:in\: AP}

  • \sf\gray{The\: greatest \:angle\: is\: double\: the\: least}

\huge\bf\red{\underline{\underline{To\:Find}}}\::

  • \sf\gray{All\: the\: angles \:in \:degrees\: \& \:radians}

\huge\bf\red{\underline{\underline{Solution}}}\::

\star\:\bf\pink{Let\:} \begin{cases}\sf\green{The \: least \: angle\: =\:x^{\circ}}\\ \sf\purple{The \: greatest \:angle\: be\:(2x)^{\circ}}\end{cases}

\to\:\:\sf\blue{x^{\circ},\:y^{\circ},\:(2x)^{\circ}}

\to\:\:\sf\purple{y\:-\:x\:=\:2x\:-\:y}

\to\:\:\sf\blue{y\:+\:y\:=\:2x\:+\:x}

\to\:\:\sf\purple{2y\:=\:3x}

\to\:\:{\underline{\boxed{\sf{\blue{y\:=\:\bigg( \dfrac{3x}{2} \bigg)^{\circ} }}}}}

\star\:\sf\underline\red{Sum\:of\:angles\:of\:a\: triangle\:=\:180^{\circ}}

\mapsto\:\:\sf\orange{x\:+\:\dfrac{3x}{2}\:+\:2x\:=\:180}

\mapsto\:\:\sf\blue{2x\:+\:3x\:4x\:=\:2\:\times\:180}

\mapsto\:\:\sf\orange{9x\:=\:360^{\circ}}

\mapsto\:\:\sf\blue{x\:=\:\dfrac{\cancel{360}}{\cancel{9}}}

\mapsto\:\:{\underline{\boxed{\sf{\orange{x\:=\:40^{\circ}  }}}}}

\mapsto\:\:\sf\blue{40\:\times\:\dfrac{\pi}{180}}

\mapsto\:\:\sf\orange{\dfrac{2\pi}{9}\:(radian)}

\star\:\sf\underline\red{2nd\:angle\:y\:=\:\dfrac{3x}{2}}

\hookrightarrow\:\:\sf\pink{\dfrac{3\:\times\:\cancel{40}}{\cancel{2}}}

\hookrightarrow\:\:\sf\green{3\:\times\:20}

\hookrightarrow\:\:\sf\pink{60^{\circ}}

\hookrightarrow\:\:\sf\green{60\:\times\:\dfrac{\pi}{180}}

\hookrightarrow\:\:\sf\pink{\dfrac{\pi}{3}\:(radian)}

\star\:\sf\underline\red{Largest\:angle\:=\:2x}

\Rightarrow\:\:\sf\purple{2\:\times\:40\:=\:80^{\circ}}

\Rightarrow\:\:\sf\green{80\:\times\:\dfrac{\pi}{180}}

\Rightarrow\:\:\sf\purple{\dfrac{4\pi}{9}\:(radian)}

Answered by Anonymous
20

 \pink{\large{\underline{\underline{ \rm{Given: }}}}}

◉ The angles are in AP.

◉ The greatest angle is double the least.

 \pink{\large{\underline{\underline{ \rm{To \: Find: }}}}}

◉ All the angles in degrees and radians.

 \pink{\large{\underline{\underline{ \rm{Let's \: Find \: Out: }}}}}

As, the required angles are in AP.

So let the required angles be ( a - d )°, a°, ( a + d )°.

We know, the sum of angles of a triangle is 180°.

So on adding all the angles, we have:

 \sf(a  - d) + a +(a  + d) = 180 \degree

 \sf a - d +  a+ a + d = 180 \degree

 \sf3a = 180 \degree

 \sf{a =  \dfrac{180 \degree}{3}}

 \sf{a = 60 \degree} .......①

It is given that the greatest angle is double the least that means the greatest angle is twice the least.

We have:

 \sf{a + d = 2(a - d)}

 \sf{a + d = 2a - 2d}

 \sf{d + 2d = 2a - a}

 \sf3d = a

We know a = 60° from

Substitute the value of a, we have:

 \sf3d = 60 \degree

 \sf{d =  \dfrac{60 \degree}{3} }

 \sf{d = 20 \degree}

Put this value in angles, we get:-

 \sf{(a - d)  \degree = 60 \degree - 20 \degree}

{ \sf{(a -d ) \degree = 40 \degree}}

 \sf{a = 60 \degree}

{ \sf{(a + d) \degree = 60 \degree + 20 \degree}}

{ \sf{(a + d) \degree = 80 \degree}}

Hence, the angles are 40°, 60°, 80° ( in degrees ).

Angles in radians :-

 \sf{(40 \times  { \dfrac{\pi}{180})   }^{c} }

  \sf=   ({ \dfrac{2\pi}{9}) }^{c}  \: or \: \dfrac{2\pi}{9}  rad

 \sf{(60 \times   { \dfrac{\pi}{180})}^{c}}

 \sf =  ({\dfrac{\pi}{3})}^{c}  \: or \:  \dfrac{\pi}{3} rad

 \sf{80 \times { \dfrac{\pi}{180} }^{c} }

 \sf = ({ \dfrac{4\pi}{9})}^{c}  \: or \:  \dfrac{4\pi}{9} rad

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