Question

The radius of a circle is 20% more than the height of art angle triangle .the base of the triangle is 36cm.if the area of triangle nd circle r equal ,what will be the area of circle

Answer 1

$\bigstar\boxed{\large\bf{\leadsto Area_{(circle)}=72\:cm^2}}$

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$\large\sf\underline{Appropriate\: question:}$⠀

❍ The radius of a circle is 20% more than the height of right angle triangle. The base of the triangle is 36cm . If the area of triangle and circle are equal ,what will be the area of circle ?

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$\large\sf\underline{ Given:}$

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• ➨ The radius of circle is 20 % more than the height of right angled triangle
• ➨ The base of triangle is 36 cm
• ➨ The area of triangle and circle are equal

$\large\sf\purple{\underline{Formulas: }}$

$\underline{\boxed{\bf{ Area_{(circle)}=\pi r^2}}}$

$\underline{\boxed{\bf{ Area_{(triangle)}=\dfrac{1}{2}\times base \times height}}}$

➠ Let the height of triangle be x cm

Then, radius of circle [( 100 +20 ) = 120 % ]

ㅤㅤㅤㅤㅤ$\sf{\longrightarrow (120\%\:of\:x)\:=\cancel{\dfrac{120}{100}}\times x}$

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ㅤㅤㅤㅤㅤ$\sf{\longrightarrow \dfrac{6x}{5}cm}$

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$\sf{\longrightarrow Area\:of\: triangle=\dfrac{1}{2}\times 36\times x }$

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$\sf{\longrightarrow Area\:of\: circle=\pi\times \left(\dfrac{6x}{5}\right)^2}$

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❍ As both the area of triangle and circle are equal , then equating the area to obtain the value of x

ㅤㅤㅤㅤㅤ$\sf{\longrightarrow \dfrac{1}{\cancel{2}}\times\cancel{36}\times x=\dfrac{22}{7}\times\dfrac{6x}{5}\times\dfrac{6x}{5}}$

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ㅤㅤㅤㅤㅤ$\sf{\longrightarrow 18\times x=\dfrac{22}{7}\times\dfrac{6x}{5}\times\dfrac{6x}{5}}$

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ㅤㅤㅤㅤㅤ$\sf{\longrightarrow \dfrac {18\times 7\times 5\times 5\times 5}{22\times 6\times 6}=\cancel\dfrac{x}{x}\times x}$

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ㅤㅤㅤㅤㅤ$\sf{\longrightarrow x=\left(\dfrac{18\times 7\times 5\times 5\times 5}{22\times 6\times 6}\right) }$

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➨ So , radius of circle :

ㅤㅤㅤㅤㅤ$\sf{\longrightarrow \dfrac{6}{5}\times x}$

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ㅤㅤㅤㅤㅤ$\sf{\longrightarrow \dfrac{6}{5}\times\left(\dfrac{18\times 7\times 5\times 5\times 5}{22\times 6\times 6}\right)}$

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ㅤㅤㅤㅤㅤ$\sf{\longrightarrow \dfrac{105}{22}cm}$

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$\large\sf\underline{ \dag Therefore,\:area\:of\: circle:}$

ㅤㅤㅤㅤㅤ$\sf{\longrightarrow\left (\dfrac{22}{7}\times\dfrac{105}{22}\times\dfrac{105}{22}\right)}$

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ㅤㅤㅤㅤㅤ$\sf{\longrightarrow\left (\dfrac{1575}{22}\right)cm^2}$

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ㅤㅤㅤㅤㅤ$\sf{\longrightarrow 71.6cm^2}$

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