
ABCD is a trapezium of area 24.5 sq.cm. In it, \[AD||BC\], $ \angle DAB = 90^\circ $, $ AD = 10cm $ and $ BC = 4cm $ . If ABE is a quadrant of a circle, find the area of the shaded region. [Take $ \pi = \dfrac{{22}}{7} $

Answer
513.6k+ views
Hint: To solve this question, here we will first derive the perpendicular AB from the area of the trapezium. Using AB as a radius of quadrant ABE we will obtain the area of the quadrant. Thus by subtracting the area of the quadrant from the area of the trapezium we will get the area of the shaded portion.
Complete step-by-step answer:
According to the question, ABCD is a trapezium.
\[AD||BC\]
$ \angle DAB = 90^\circ $
$ AD = 10cm $
And, $ BC = 4cm $
ABE is a quadrant of a circle.
Area of the trapezium is 24.5 sq.cm.
We know that the area of the trapezium is half of the sum of parallel sides multiplied by perpendicular distance between them.
i.e. area of the trapezium = $ \dfrac{1}{2} $ (sum of parallel sides)(perpendicular)
Area of the trapezium ABCD $ = \dfrac{1}{2} \times (AD + BC) \times AB $
Putting the value of area of trapezium and AD and BC in above equation we get,
$ 24.5 = \dfrac{1}{2} \times (10 + 4) \times AB $
Simplifying the above equation we get,
$ 24.5 = 7AB $
$ \Rightarrow AB = 3.5 $
It is also given that ABE is a quadrant of a circle.
i.e. ABE is the one fourth of a circle. And its radius is AB.
The angle of ABE is given by, $ \angle DAB = 90^\circ $
Hence, the area of the quadrant ABE $ = \dfrac{\theta }{{360^\circ }}\pi {(AB)^2} $
Putting the values in the above equation we get,
The area of the quadrant ABE $ = \dfrac{{90^\circ }}{{360^\circ }} \times \dfrac{{22}}{7}{(3.5)^2} $
$ = \dfrac{1}{4} \times \dfrac{{22}}{7} \times 12.25 = 9.625c{m^2} $
From the figure, Area of the shaded portion = Area of the trapezium – Area of the quadrant ABE
$ = 24.5 - 9.625 $
$ = 14.875c{m^2} $
$ \therefore $ The area of the shaded portion is $ 14.875c{m^2} $
Note: Trapezium is a convex quadrilateral with one pair of parallel sides and one pair of non-parallel sides.
Area of the trapezium is given by product of the half of the sum of the parallel sides and perpendicular distance.
Quadrant of a circle is one fourth of a circle that involves angle $ 90^\circ $ .
Area of the quadrant of a circle $ = \dfrac{\theta }{{360^\circ }}\pi {r^2} $, where r is the radius.
You should remember all the formulas and properties of geometric shapes.
Complete step-by-step answer:
According to the question, ABCD is a trapezium.
\[AD||BC\]
$ \angle DAB = 90^\circ $
$ AD = 10cm $
And, $ BC = 4cm $
ABE is a quadrant of a circle.
Area of the trapezium is 24.5 sq.cm.
We know that the area of the trapezium is half of the sum of parallel sides multiplied by perpendicular distance between them.
i.e. area of the trapezium = $ \dfrac{1}{2} $ (sum of parallel sides)(perpendicular)
Area of the trapezium ABCD $ = \dfrac{1}{2} \times (AD + BC) \times AB $
Putting the value of area of trapezium and AD and BC in above equation we get,
$ 24.5 = \dfrac{1}{2} \times (10 + 4) \times AB $
Simplifying the above equation we get,
$ 24.5 = 7AB $
$ \Rightarrow AB = 3.5 $
It is also given that ABE is a quadrant of a circle.
i.e. ABE is the one fourth of a circle. And its radius is AB.
The angle of ABE is given by, $ \angle DAB = 90^\circ $
Hence, the area of the quadrant ABE $ = \dfrac{\theta }{{360^\circ }}\pi {(AB)^2} $
Putting the values in the above equation we get,
The area of the quadrant ABE $ = \dfrac{{90^\circ }}{{360^\circ }} \times \dfrac{{22}}{7}{(3.5)^2} $
$ = \dfrac{1}{4} \times \dfrac{{22}}{7} \times 12.25 = 9.625c{m^2} $
From the figure, Area of the shaded portion = Area of the trapezium – Area of the quadrant ABE
$ = 24.5 - 9.625 $
$ = 14.875c{m^2} $
$ \therefore $ The area of the shaded portion is $ 14.875c{m^2} $
Note: Trapezium is a convex quadrilateral with one pair of parallel sides and one pair of non-parallel sides.
Area of the trapezium is given by product of the half of the sum of the parallel sides and perpendicular distance.
Quadrant of a circle is one fourth of a circle that involves angle $ 90^\circ $ .
Area of the quadrant of a circle $ = \dfrac{\theta }{{360^\circ }}\pi {r^2} $, where r is the radius.
You should remember all the formulas and properties of geometric shapes.
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