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In the given figure, $ABCD$ is a square of side $5 cm$ inscribed in a circle. What is the area of the shaded region[Take $π = 3.14$ ] ?

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14.25 c m^{2}

16.75 c m^{2}

25 c m^{2}

15.25 c m^{2}

answer is A.

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Detailed Solution

Given,

ABCD

is a square of side

5 cm

inscribed in a circle.
Question Image

Diagonal

AC

is the diameter of the circle with centre

O

∴ OC = OA =

radius of the circle.

AB = AC = 5 cm

△ ABC

is a right-angled triangle.
Pythagoras theorem states that the square of the hypotenuse side in a right-angled triangle equals the sum of the squares of the other two sides.
According to Pythagoras's theorem,

A C^{2} = A B^{2} + A C^{2}

\Rightarrow

A C^{2} = 5^{2} + 5^{2}

\Rightarrow

A C^{2} = 50 cm

\Rightarrow AC = 5 \sqrt{2}

∴ OA = OC = \frac{AC}{2}

.
Radius

= \frac{5 \sqrt{2}}{2}

Area of a square is

sid e^{2}

\Rightarrow

Area of square

= A C^{2}

\Rightarrow

Area of square

= 5^{2}

\Rightarrow

Area of square

= 25 c m^{2}

Area of circle

= π r^{2}

\Rightarrow

Area of circle

= 3.14 \times {(\frac{5 \sqrt{2}}{2})}^{2}

\Rightarrow

Area of circle

= 3.14 \times \frac{50}{4}

\Rightarrow

Area of circle

= 39.25 c m^{2}

∴

Area of the shaded region

=

Area of circle

-

Area of square.

\Rightarrow

Area of the shaded region

= 39.25 - 25

\Rightarrow

Area of the shaded region

= 14.25 c m^{2}

Hence, option 1 is correct.

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In the given figure, ABCD is a square of side 5 cm inscribed in a circle. What is the area of the shaded region[Take π=3.14] ?

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In the given figure, $ABCD$ is a square of side $5 cm$ inscribed in a circle. What is the area of the shaded region[Take $π = 3.14$ ] ?