Question

15. In the given figure, ABCD is a square of side 5 cm
inscribed in a circle. Find (i) the radius of the circle (ii) the
area of the shaded region. (Take 1 = 3.14)​

Answer 1

$\huge \sf{ \boxed{ \underline{ \blue{ \sf{ = 14.25 \: }}}}}$

$\sf \huge \underline{Question}$

In the given figure, ABCD is a square of side 5 cm inscribed in a circle. Find (i) the radius of the circle (ii) the area of the shaded region. (Take 1 = 3.14)

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$\sf \huge \underline \red{To \: find}$

• radius of the circle

• the area of the shaded region

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$\sf \huge\underline \pink{Answer}$

• Ac is the diameter of the circle center 0

• OC = OA = the radius of the circle

$\tt \blue{given}$

• ABCD is a square of a side is 5cm

$\rm \implies \green{AB = AC= 5cm}$

$\tt \implies \red{ABC \: is \: right \: angled \: traingle}$

therefore,

$\rm \implies \pink{{ AC}^{2} =A {B}^{2} + A {C}^{2}}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm \implies \blue{ = {5}^{2} + {5}^{2}}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm \implies \purple{ = 25 + 25}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm \implies \orange{ = 50cm}$

then,

$\tt \implies \red{AC = 5 \sqrt{2}}$

$\bf \underline{now \: radius \: of \: circle}$

$\: \: \: \: \: \: \: \: \bf \implies\pink{OA = OC = \dfrac{ AC\: }{2}}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bf \implies \blue{R= \dfrac{5 \sqrt{2} }{2}}$

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$\tt \implies \orange{area \: of \: square = A {C}^{2}}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt \orange{ = {5}^{2}}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt \orange{ = 25}$

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$\tt \implies \red{area \:of \: circle = \pi {r}^{2}}$

• from Given Question

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \ \: \: \: \: \: \: \: \: \tt \red{ = 3.14 \times (\dfrac{5 \sqrt{2} }{2} ) {}^{2}}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt \red{ = 3.14 \times \dfrac{50}{4}}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt \red{ = 39.25}$

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• Now use the area of the shaded region formula

Area of shaded region = Area of circle = area of square

• area of circle is 39.25

• area of square is 25

according to this formula

$\rm \orange{ = 39.25 - 25}$

$\rm \orange{ = 14.25 {cm}^{2}}$

Answer 2

Answer ▬

$hlo$

▬▬▬▬▬▬▬❤❤❤

❤Ac is the diameter of the circle center 0

❤OC = OA = the radius of the circle

\given

ABCD is a square of a side is 5cm

$</p><p></p><p>\rm \implies \green{AB = AC= 5cm}⟹AB=AC=5cm</p><p></p><p>\tt \implies \red{ABC \: is \: right \: angled \: traingle}⟹ABCisrightangledtraingle</p><p></p><p></p><p></p><p>\rm \implies \pink{{ AC}^{2} =A {B}^{2} + A {C}^{2}}⟹AC2=AB2+AC2</p><p></p><p>\: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm \implies \blue{ = {5}^{2} + {5}^{2}}⟹=52+52</p><p></p><p>\: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm \implies \purple{ = 25 + 25}⟹=25+25</p><p></p><p>\: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm \implies \orange{ = 50cm}⟹=50cm</p><p></p><p></p><p></p><p>\tt \implies \red{AC = 5 \sqrt{2}}⟹AC=52</p><p></p><p>$

$</p><p></p><p>\: \: \: \: \: \: \: \: \bf \implies\pink{OA = OC = \dfrac{ AC\: }{2}}⟹OA=OC=2AC</p><p></p><p>\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bf \implies \blue{R= \dfrac{5 \sqrt{2} }{2}}⟹R=252</p><p></p><p>________________________________________</p><p></p><p>\tt \implies \orange{area \: of \: square = A {C}^{2}}⟹areaofsquare=AC2</p><p></p><p>\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt \orange{ = {5}^{2}}=52</p><p></p><p>\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt \orange{ = 25}=25</p><p></p><p>_________________________________________</p><p></p><p>$

$from Given Question</p><p></p><p>\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \ \: \: \: \: \: \: \: \: \tt \red{ = 3.14 \times (\dfrac{5 \sqrt{2} }{2} ) {}^{2}} =3.14×(252)2</p><p></p><p>\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt \red{ = 3.14 \times \dfrac{50}{4}}=3.14×450</p><p></p><p>\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt \red{ = 39.25}=39.25</p><p></p><p>$

❤Now use the area of the shaded region formula

Area of shaded region = Area of circle = area of square

✨area of circle is 39.25

✨area of square is 25

✨according to this formula

14.25cm²