Question

18 cm
6. In the given figure, ABCD is a quadrilateral in which AB | CD and CD I DA. IF AB
DC = BC = 10 cm, find the area of the quadrilateral ABCD.
10 cm D
K10 cm
В.
16 cm
A A​

Answer 1

Answer:

Answer

Let CD=x then AB=2CD=2x.

Let r be the radius of the circle inscribed in the quadrilateral ABCD.

ar(□ABCD)=18 and AB∥CD.

⇒

2

1

(x+2x).2r=18

⇒ 3xr=18

⇒ xr=6 ----- ( 1 )

OP=OM=PD=OQ=AM=r

⇒ PC=x−r and MB=2x−r

Let ∠PCO=∠OCQ=θ

In right-angled △OPC,

tanθ=

CP

OP

=

x−r

r

----- ( 2 )

CD∥AB

∴ ∠PCB=∠QOM=2θ

⇒ ∠CBA=180

o

−2θ

⇒ ∠OBM=90

o

−θ

From △OMB,

tan(90

o

−θ)=

MB

OM

=

2x−r

r

Right-angled △OBM,

⇒ tanθ=

r

2x−r

------ ( 3 )

From ( 2 ) and ( 3 ),

⇒

x−r

r

=

r

2x−r

⇒ 2x

2

−3xr=0

⇒ x(2x−3r)=0

⇒ x=

2

3r

---- ( 4 )

From ( 1 ) and ( 4 ), we get

⇒ xr=6

⇒

2

3rr

=6

⇒ r

2

=4

⇒ r=2

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