Question

a trapezium of parellel sides 10cm and 21cm and height 8cm is inscribed in a circle of radius 7cm. calculate the area of the region not occupied by the trapezium

Answer 1

$\huge\mathfrak{Answer:}$

$\large\underline{\sf{\blue{Given:}}}$

• We have been given a Trapezium of parallel sides 10 cm and 21 cm and height 8 cm
• The Trapeziums is inscribed in a circle of radius 7 cm

$\large\underline{\sf{\blue{To \: Find:}}}$

• We have to find the area of region not occupied by the Trapezium

$\large\underline{\sf{\blue{Concept \: Used:}}}$

Circle:

Circle is the locus of all the points equidistant from a given fixed point . The fixed point is known as center

$\boxed{\sf{Area \: of \: circle = \pi \: r^2}}$

$\sf{}$

Trapezium:

Trapezium is a Quadrilateral having two opposite sides parallel and two sides non parallel

$\boxed{\sf{Area = \dfrac{1}{2} \times (Sum \: of \: Parallel \: Sides) \times h }}$

$\large\underline{\sf{\blue{Solution:}}}$

On Analyzing the Question figure can be drawn as shown in [ Attachment ]

Let ABCD be the Trapezium inscribed in a circle having center O

$\sf{}$

$\odot \: \: \underline{\sf{\orange{Area \: of \: Given \: Circle :}}}$

$\large\boxed{\sf{Radius \: of \: circle = 7 cm}}$

Area of circle can easily be determined by using the given formula

$\implies \sf{Area_{(circle)} = \pi \: r^2}$

$\implies \sf{Area_{(circle)} = \dfrac{22}{7} \times 7 \times 7}$

$\implies \boxed{\sf{Area_{(circle)} = 154 \: cm^2}}$

$\sf{}$

$\odot \: \: \underline{\sf{\orange{Area \: of \: Given \: Trapezium :}}}$

First Parallel side $\sf{(P_1)}$ = 10 cm

Second Parallel side $\sf{(P_2)}$ = 21 cm

Height of Trapezium ( h ) = 8 cm

Area of Trapezium can be determined by using the following formula

$\implies \sf{Area_{(Trapezium)} = \dfrac{1}{2} \times (P_1 + P_2) \times h}$

$\implies \sf{Area_{(Trapezium)} = \dfrac{1}{2} \times (10 + 21) \times 8}$

$\implies \sf{Area_{(Trapezium)} = 31 \times 4}$

$\implies \large\boxed{\sf{Area_{(Trapezium)} = 124 \: cm^2}}$

$\sf{}$

$\odot \: \: \underline{\sf{\orange{Required \: Area :}}}$

Area of circle not occupied by the Trapezium can be determined by

$\sf{Required \: Area =Area_{(circle)} - Area_{(Trapezium)}}$

$\sf{Required \: Area = 154 - 124}$

$\boxed{\sf{Required \: Area = 30 \: cm^2}}$

Hence Area of circle not occupied by the Trapezium is 30 $\sf{cm^2}$

$\sf{}$

$\mathtt{\pink{Answer . . .}}$

Answer 2

Answer:

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