The rational number which equals the number `2. 357 ` with recurring decimal is

Options

A. `(25)/(9)`
B. `(25)/(3)`
C. `(25)/(18)`
D. `(10)/(3)`

Explanation

Let G be the point of intersection of the medians of triangle ABC. Then the area of `DeltaABC` is three times that of `DeltaAGC`
Now, in `DeltaAGC, AG = (2)/(3) AD = (10)/(3)`
Applying the sine rule of `DeltaAGC`, we get
`(GC)/(sin(pi//8)) = (AG)/(sin(pi//4)) rArr GC = (10)/(3) (sin(pi//8))/(sin(pi//4))`
Area of `DeltaAGC = (1)/(2) AG GC sin angle AGC`
`=(1)/(2) (10)/(3) (10)/(3) (sin(pi//8))/(sin(pi//4)) sin ((pi)/(2) + (pi)/(8))`
`= (50)/(9) (sin (pi//8) cos (pi//8))/(sin(pi//4))`
`= (50)/(18) = (25)/(9)`
`:.` Area of `DeltaABC = 3 xx (25)/(9) = (25)/(3)`