Question

Square $ABCD$ has area $200$. Point $E$ lies on side $\overline{BC}$. Points $F$ and $G$ are the midpoints of $\overline{AE}$ and $\overline{DE}$, respectively. Given that quadrilateral $BEGF$ has area $34$, what is the area of triangle $GCD$?

Answer 1

1. Consider square ABCD. You know that

`A_(ABCD)=AD^2=200,`

then

`AB=BC=CD=AD=10√(2).`

2. Consider traiangle AED. F is mipoint of AE and G is midpoint of DE, then FG is midline of triangle AED. This means that

`FG=(AD)/(2)=(10√(2) )/(2)=5√(2).`

3. Consider trapezoid BFGC. Its area is

`A_(BFGC)=(FG+BC)/(2)\cdot h,` where h is the height of trapezoid and is equal to half of AB. Thus,

`A_(BFGC)=(FG+BC)/(2)\cdot (AB)/(2)=(5√(2)+10√(2))/(2)\cdot (10√(2))/(2)=75.`

4.

`A_(BFGC)=A_(BFGE)+A_(EGC),\\A_(EGC)=A_(BFGC)-A_(BFGE)=75-34=41.`

5. Note that angles EGC and CGD are supplementary and

`\sin \angle CGD=\sin \angle EGC.`

Then

`A_(CGD)=(1)/(2)CG\cdot CD\cdot \sin \angle CGD=(1)/(2)CG\cdot EG\cdot \sin \angle CGE=A_(ACG)=41.`

Answer:
`A_(CGD)=41.`