Question

let ABCD be a quadrilateral with area 18 with side AB parallel to side CD and AB =2CD. let AD be perpendicular to AB and CD .if a circle is drawn inside the quadrilateral ABCD touching all the sides then it's radius is ?

Answer 1

Let CD=xCD=x then AB=2CD=2xAB=2CD=2x.Let rr be the radius of the circle inscribed in the quadrilateral ABCD.

Given: Area of quadrilateral ABCD=1818and ABis||ABis|| to CDCD.

⇒12⇒12(x+2x).2r=18(x+2x).2r=18

⇒3xr=18⇒3xr=18

⇒xr=6⇒xr=6-----(1)

OP=OM=PD=OQ=AM=rOP=OM=PD=OQ=AM=r

⇒PC=x−randMB=2x−r⇒PC=x−randMB=2x−r

Let ∠PCO=angleOCQ=θ∠PCO=angleOCQ=θ then from right-angled ΔOPCΔOPC

tanθ=OPCP=rx−rtan⁡θ=OPCP=rx−r-----(2)

CD∥ABCD∥AB

∴∠PCB=∠QOM=2θ∴∠PCB=∠QOM=2θ

Step 2:

∠CBA=180∘−2θ∠CBA=180∘−2θ

∠OBM=90∘−θ∠OBM=90∘−θ

⇒⇒ From ΔOMB,tan(90∘−θ)=OMMB=r2x−rΔOMB,tan⁡(90∘−θ)=OMMB=r2x−r

From right angled ΔOBMΔOBM

tanθ=2x−rrtan⁡θ=2x−rr-----(3)

From (2) & (3)

rx−r=2x−rrrx−r=2x−rr

⇒2x2−3xr=0⇒2x2−3xr=0

x(2x−3r)=0x(2x−3r)=0

x=3r2x=3r2-------(4)

From (1) & (4) we get,

xr=6xr=6

3rr2=3rr2=66

r2=4r2=4

r=2