Question

Question #00f2d

Answer 1

`r=sqrt(7/3)`

Explanation:

I assume that `AB=BC=CD=1, and DE=EF=FA=2`

As shown in the diagram,
`3(x+y)=360^@ => x+y=120^@`
`w=(180-x)/2, z=(180-y)/2`
`=> w+z=(360-(x+y))/2=(360-120)/2=120^@`

In `DeltaCDE`,
`CE^2=CD^2+DE^2-2(CD)(DE)cos(w+z)`
`CE^2=1^2+2^2-2*1*2*cos120=7`

In `DeltaOCE`,
`CE^2=OC^2+OE^2-2(OC)(OE)cos(x+y)`
`7=r^2+r^2-2r^2cos120`
`=> 7=2r^2(1-cos120)=2r^2(1-(-1/2))=3r^2`
`=> r^2=7/3`
`=> r=sqrt(7/3)`