Question

From a thin metallic piece, in the shape of a trapezium ABCD, in which AB || CD and ∠BCD = 90°, a quarter circle BEFC is removed (in the following figure). Given AB = BC = 3.5 cm and DE = 2 cm, calculate the area of the remaining piece of the metal sheet.​

Answer 1

Answer:

The area of remaining piece of the metal sheet (Shaded region) is 6.125 cm².

Step-by-step explanation:

Given :

In trapezium ABCD in which AB || CD and ∠BCD = 90°  

AB = BC =3.5 cm & DE = 2 cm  

 

CE = CB = 3.5 cm  

[CE and BC are the radii of quarter circle BFEC]

so,DC = DE + EC  

DC = 2 cm + 3.5 cm  

DC = 5.5 cm  

Area of remaining piece of the metal sheet (Shaded region), A = Area of trapezium ABCD - Area of quarter circle BFEC  

= ½ (AB + DC) × BC - ¼ π(BC)²

[Area of trapezium = ½ (sum of parallel sides) × perpendicualr distance between the parallel sides & area of quadrant = ¼ π r²]

A = ½ (3.5 + 5.5) × 3.5  - ¼ π(3.5)²

A = ½ × 9  × 3.5 - ¼ π(3.5)²

A = 4.5 × 3.5  - 22/7 × 3.5 × 3.5/4

A = 15.75 - ½ × 11 × 0.5 × 3.5

A = 15.75 - 19.25/2

A = 15.75 - 9.625

A = 6.125 cm²  

Area of remaining piece of the metal sheet (Shaded region) = 6.125 cm²

Hence, the area of remaining piece of the metal sheet (Shaded region) is 6.125 cm².

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Answer 2

$\huge\underline\mathfrak\purple {Solution }$

It is given that ABCD is a trapezium with AB || CD and <BCD = 90° .

And

AB = BC = 3.5 cm

and DE = 2 cm

=> CE = BC = 3.5 cm

( CE and BC are the radii of the same quarter circle BFEC )

Area of shaded region = ??

Now,

DC = DE + EC

DC = 2+3.5

DC = 5.5 cm

Area of shaded region = area of trapezium ABCD - Area of quarter circle BEFC

$= \frac{1}{2} \times (ab + dc) \times bc - ( \frac{1}{4} \pi {bc}^{2} )$

$= \frac{1}{2} \times (3.5 + 5.5) \times 3.5 - ( \frac{1}{4} \times \frac{22}{7} \times ( {3.5}^{2}))$

$= \frac{1}{2} \times 9 \times 3.5 - ( \frac{1}{2} \times \frac{11}{7} \times 12.25)$

$= \frac{1}{2} \times 31.5 - ( \frac{1}{2} \times 11 \times 1.75)$

$= 15.75 - (5.5 \times 1.75)$

$= 15.75 - 9.625$

$= 6.125$

Hence , the area of the shaded region is 6.125 cm^2 .