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MisterBrian

Answer:

Area(Shaded) = 32π-64 cm²

Perimeter (Shaded) = 8π cm

Step-by-step explanation:

Area of quarter circle ADC: 1/4*π*radius²

Area of square : side²,here side = 8 cm

Here the radius is 8 cm

  • Ar(Quat.)=1/4*π*r² = 1/4*π*8²

= 1/4*8*8*π = 16π

Area of region ABC = Area(Square) - Area(Quat.)

  • Ar(ABC) = 8² - 16π = 64-16π

Similarly the area of quarter circle ACB = 16π (Same as above)

Similarly area of region ADC: Area(Square) - Area(Quat.)

  • Ar(ADC) = 8²-16π = 64-16π

Now

  • Ar(ADC+ABC) = 64-16π+64 - 16π

= (64+64)-(16π+16π)

= 128 - 32π

Now Area of shaded region AC:

  • Ar(Shaded) = Area(Sq.) - Ar(ADC+ABC)

= 8²-(128-32π) = 64-128+32π

= 32π - 64 cm²

(ii) Perimeter of shaded:

For quarter circle ADC,the arc length AC is

  • 1/4(2πr) = 1/2(πr) = 1/2(π)(8) = 4π

Similarly the arc length of AC of quarter circle ACB

  • 1/4(2πr) = 1/2(πr) = 1/2(π)(8) = 4π

Adding Arc AC + Arc AC = 4π+4π = 8π , we get this as perimeter since as it's definition states the only length of boundary.

semsee45

Answer:

[tex]A = \boxed{32\pi - 64}\; \sf cm^2[/tex]

[tex]P = \boxed{8\pi}\;\sf cm[/tex]

Step-by-step explanation:

Area

The given diagram shows a square ABCD with two unshaded white areas. Each of these unshaded white areas is the result of subtracting the area of a quarter circle with radius r from the area of a square with side length r.

The formula for the area of a quarter circle with radius r is:

[tex]\textsf{Area of a quarter circle}=\dfrac{\pi r^2}{4}[/tex]

Therefore, the area of the entire unshaded white region is:

[tex]\begin{aligned}\textsf{Area of white region}&=2 \times (\textsf{Area of a square}-\textsf{Area of a quarter circle})\\\\&=2 \times \left(r^2-\dfrac{\pi r^2}{4}\right)\\\\&=2r^2-\dfrac{\pi r^2}{2}\end{aligned}[/tex]

To find the area of the shaded region, we need to subtract the area of the white unshaded region from the area of the square:

[tex]\textsf{Area of the shaded region}=r^2-\left(2r^2-\dfrac{\pi r^2}{2}\right)[/tex]

Given the side length of the square is 8 cm, we can substitute r = 8 into the formula:

[tex]\begin{aligned}\textsf{Area of the shaded region}&=8^2-\left(2\cdot 8^2-\dfrac{\pi \cdot 8^2}{2}\right)\\\\&=64-\left(2\cdot 64-\dfrac{64\pi}{2}\right)\\\\&=64-\left(128-32\pi\right)\\\\&=64-128+32\pi\\\\&=32\pi-64\end{aligned}[/tex]

Therefore, the area of the shaded region is (32π - 64) cm².

[tex]\hrulefill[/tex]

Perimeter

The perimeter of the shaded region is made up of two curved edges, each belonging to a quarter circle with radius r. Since the curved edge of a quarter circle is a quarter of the circumference of a whole circle, the perimeter of the shaded region is equal to half the circumference of a circle.

The expression for the circumference of a circle with radius r is 2πr. Therefore, the expression for half a circumference is πr, so:

[tex]\textsf{Perimeter of shaded region}=\pi r[/tex]

Given that the radius of the quarter circle is equal to the side length of the square (8 cm), we can substitute r = 8 into the formula:

[tex]\textsf{Perimeter of shaded region}=8\pi[/tex]

Therefore, the perimeter of the shaded region is 8π cm.

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