ABCD is a parallelogram, G is the point on AB such that AG = 2GB, E is a point of DC , such that CE = 2DE and F is the point of BC such that BF = 2FC. prove that :

1) ar (quad ADEG) = ar(quad GBCE)

2.) ar( triangle EBG) = 1/6 ar( quad ABCD)

3) ar( triangle EFC) = 1/2 AR(triangle EBF)

4 ) ar( triangle EBG )= ar( triangle EFC)

5 ) find what portion of the area of parallelogram is the area of triangle EFG.

"meritnation expert gursheen kaur has answered my question, but the method was incorrect.she supposed that G is the mid-point of AB and F is the midpoint of BC. but it's wrong method. please experts firstly read the question properly before answering it."

given: ABCD is a parallelogram. AG=2GB,CE=2DE and BF=2FC.

(i)

TPT: area(quad ADEG)=area(quad GBCE)

construction: from point E and G draw the  lines intersecting AB at N and DC at M respectively.

proof: since ABCD is a parallelogram. AB=CD

BG = 1/3 AB and DE=1/3 DC=1/3 AB

therefore DE=BG

here ADEN is a parallelogram [since AN is parallel to DE and AD is parallel to NE]

quadrilateral BCMG is also the parallelogram.

area (parallelogram ADEN)=area(parallelogram BCMG )............(1)  [since DE=BG and AD=BC parallelogram with corresponding sides equal]

area(Δ NEG)= area(ΔEGM)..........(2) [diagonals of a parallelogram divide it into two equal areas]

adding (1) and (2)

area (parallelogram ADEN)+area(ΔNEG)=area(parallelogram BCMG )+area(ΔEGM)

⇒area(quad ADEG)=area(quad GBCE)

therefore (i) part is proved

(ii)

TPT: area(ΔEBG)=1/6 area(ABCD)

proof:

let the distance between the parallel lines AB and CD be h.

therefore area (ABCD)= AB*h............(1)

since AG=2BG, and AG+BG=AB

therefore BG=1/3 AB

area of (ΔEBG)=............(2)

from (1) and (2), we have area(ΔEBG)=1/6 area(ABCD)

therefore (ii) part is proved.

(iii)

TPT: area(ΔEFC)=1/2area(ΔEBF)

proof:

let the distance between the parallel sides EN and CB be x.

therefore area(ΔBEF)=.............(1)

area (ΔECF)=......(2)

which is (iii) result.

now  try to solve the (iv) and (v) part by yourself , if you face any difficulty then do get back to us.