Question

The figure below shows a square ABCD of side 6 cm. Given that E is the midpoint of AB, points F and G are on BC so that BF = FG = GC. What is the total area of the shaded region in `cm^2`?

Answer 1

`"shaded area" = 6.3 cm^2`

Explanation:

There should be other better methods.

Total area of the shaded region `A_S=x+2y-z`,
where `x, y and z` are areas of the respective region as shown in Fig 1 and Fig 2.
Let `|PQR|` denote Area of `DeltaPQR`

See Fig 1 :
a) find `x`
As `DeltaEJA and DeltaHJA` are congruent.
Let `|EJA| =|HJA|=x, => |DJH|=x`
`=> |DEA|=x+x+x=3x=1/2xx3xx6=9`
`=> x=3 cm^2`

b) find `y`
Let `|DKL|=y, => |LKC|=2y`
As `DeltaLKC and DeltaFKC` are congruent,
`=> |LKC|=|FKC|=2y`
`=> |DCF|=y+2y+2y=5y=1/2xx4xx6=12`
`=> y=12/5=2.4 cm^2`

c) find `z`
See Fig 2:
As `DeltaGNC and DeltaMNC` are congruent,
Let `|GNC |= |MNC|=z, => |MND|=2z`
`=> |DCG|=2z+z+z=4z=1/2xx2xx6=6`
`=> z=6/4=1.5 cm^2`

`=>` Area `FKNG = |FKC|-|GNC|=2y-z`

`=> "shaded area " A_S=|EJA| + "Area " FKNG`
`=x+2y-z=3+2xx2.4-1.5=6.3 cm^2`