Question

KLMN is a quadrilateral in which E,F,G and H are mid points of the sides KL,LM,MN and NK . KMis a diagonal show that i) EF is parallel to KM and EF=1/2 KM
ii)EF=HG
iii)EFGH is a paralellogram

Answer 1

Here, we are joining K and M.
In ΔKLM
E is the mid point of KL
F is the mid point of LM
EF∣∣KM [Line segments joining the mid points of two sides of a triangle is parallel to KM(third side) and also is half of it]
EF= 1/2 KM
In ΔKNM
G is mid point of MN
H is mid point of KN
GH∣∣KM [Line segments joining the mid points of two sides of a triangle is parallel to third side and also is half of it]
GH= 1/2 KM
So, EF∣∣GH and EF=GH [one pair of opposite side is parallel and equal]
In ΔKEH & ΔLEF
KE=LE [E is the mid point of KL)
∠EKH=∠ELF(All the angles of rectangle are 90
o
)
KH=LF
∴ΔKEH≅ΔLEF(SAS congruency)
∴EH=EF
LH=EF & EF=GH (opposite sides of parallelogram is equal)
∴ EF=FG=GH=EH[All sides are equal]
∴ EFGH is a parallelogram with all sides equal