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This is the second of our many quizzes on Circle arcs. Click on this to access more such competitive quizzes on Geometry.

Circle arcs Quiz 2

mathematics

 Find the length of the minor arc if the area of the region bounded by it is $576c{m^2}$, given that the minor segment is a quadrant of circle. $ \\ $<img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/db7d0221-e012-41bc-95fd-9a9666ca6a9c4776196482690884949.png"/>

 Area of segment is given as $\dfrac{\theta }{{360}}\pi {r^2}$. 

  $  {\text{Given, area of minor segment}} = 576cm  \\   \Rightarrow \dfrac{\theta }{{360}}\pi {r^2} = 576  \\   \Rightarrow \dfrac{{90}}{{360}} \times \dfrac{{22}}{7} \times {r^2} = 576  \\   \Rightarrow r = \dfrac{{\sqrt {8064} }}{{11}} = 27.07\;cm  \\  \therefore {\text{Length of arc}} = \dfrac{{90}}{{360}}2\pi r  \\   \Rightarrow \dfrac{1}{4} \times 2 \times \dfrac{{22}}{7} \times 27.07  \\   \Rightarrow 42.55\;cm  \\  $ 

 Find the ratio of the two lengths of Arc if their respective radii are $21cm$ and $54cm$ and the  angles subtended at  center is $35^\circ $ and $75^\circ $ respectively

 Apply the formula of arc length $\dfrac{\theta }{{360}}2\pi r$ to get the ratio.

  $  {\text{Given, }}{r_1} = 21cm,{r_2} = 54cm,{\theta _1} = 35^\circ ,{\theta _2} = 75^\circ   \\  {\text{Ratio}} = \dfrac{{{\text{Length of Ar}}{{\text{c}}_{\text{1}}}}}{{{\text{Length of Ar}}{{\text{c}}_{\text{2}}}}}  \\   \Rightarrow \dfrac{{\dfrac{{{\theta _1}}}{{360}}2\pi {r_1}}}{{\dfrac{{{\theta _2}}}{{360}}2\pi {r_2}}} = \dfrac{{35 \times 21}}{{75 \times 54}}  \\   \Rightarrow \dfrac{{49}}{{270}}  \\  \therefore {\text{Ratio}} = 49:270  \\  $

 The ratio of area of square to the area of region bounded by a shaded quadrant is $4$. If the side of the square is $4cm$, then find the length of the arc. $ \\ $<img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/964097c2-b404-45aa-b18a-db823f3559f78829385485305280849.png"/>

 Find the radius of the shaded quadrant by forming the relation between area of square and area of quadrant.

$  {\text{Consider the given,}}  \\  {\text{Area of square}} = 4 \times {\text{Area of quadrant}}  \\   \Rightarrow {\text{16  =  4}} \times \dfrac{\theta }{{360^\circ }}\pi {r^2}  \\   \Rightarrow 16 = 4 \times \dfrac{1}{4} \times 3.14 \times {r^2}  \\   \Rightarrow \dfrac{{16}}{{3.14}} = {r^2}  \\  \therefore {\text{Length}} = \dfrac{\theta }{{360^\circ }}2\pi r  \\   \Rightarrow \dfrac{{90^\circ }}{{360^\circ }} \times 2 \times 3.14 \times 2.26  \\   \Rightarrow 3.55cm  \\  $

 Find the ratio of difference in arc lengths to the difference in segment area of quadrants with radius $10cm$ and $8cm$.

 Use arc length and area of minor segment formula and then further solve to get the ratio. 

 $  {\text{Given, }}{r_1} = 10cm,{r_2} = 8cm  \\  {\text{Change of length }} = \dfrac{\theta }{{360^\circ }}2\pi {r_1} - \dfrac{\theta }{{360^\circ }}2\pi {r_2}  \\   \Rightarrow \dfrac{\theta }{{180^\circ }}\pi \left( {10 - 8} \right)  \\   \Rightarrow \dfrac{1}{2}\pi  \times 2 = \pi   \\  {\text{Difference between area of minor segment,}}  \\   \Rightarrow \dfrac{\theta }{{360^\circ }}\pi r_1^2 - \dfrac{\theta }{{360^\circ }}\pi r_2^2  \\   \Rightarrow \dfrac{{90^\circ }}{{360^\circ }} \times \pi \left( {100 - 64} \right)  \\   \Rightarrow \dfrac{1}{4}\pi  \times 36  \\   \Rightarrow 9\pi   \\  \therefore {\text{Ratio}} = \dfrac{\pi }{{9\pi }} = 1:9  \\  $

 If the length of an arc is $66cm$ and the area of the minor segment is $616c{m^2}$ then find the reflexive angle of center.

 Reflexive angle is greater than $180^\circ $ and less than $360^\circ $.

 $  {\text{Given, Length of arc}} = 44cm  \\  \dfrac{\theta }{{360^\circ }}2\pi r = 44  \\   \Rightarrow \pi r = \dfrac{{44 \times 360^\circ }}{{\theta  \times 2}}  \\   \Rightarrow \pi r = \dfrac{{22 \times 360^\circ }}{\theta }{\text{     }}..\left( i \right)  \\  {\text{Area of minor segment}} = 462  \\   \Rightarrow \dfrac{\theta }{{360^\circ }} \times \pi {r^2} = 462  \\   \Rightarrow \dfrac{\theta }{{360^\circ }} \times \pi r \times r = 462  \\   \Rightarrow \dfrac{\theta }{{360^\circ }} \times \dfrac{{22 \times 360^\circ }}{\theta } \times r = 462  \\   \Rightarrow r = \dfrac{{462}}{{22}}  \\   \Rightarrow r = 21cm{\text{     }}...\left( {ii} \right)  \\  {\text{Use in }}\left( i \right)  \\   \Rightarrow \dfrac{{22}}{7} \times 21 = \dfrac{{22 \times 360^\circ }}{\theta }  \\   \Rightarrow \theta  = \dfrac{{22 \times 360^\circ  \times 7}}{{21 \times 22}}  \\   \Rightarrow \theta  = 120^\circ   \\  \therefore {\text{Reflexive angle}} = 360^\circ  - 120^\circ  = 240^\circ   \\  $