Question

PQRS is a parallelogram. X and Y are midpoints of sides PQ and RS respectively. W and Z are points of intersection of SX and PY and XR and YQ respectively. show that ar(YWZ)=ar(XWZ)

Answer 1

Construction- Join WZ.
Now,
wx=WS------(1). [W is intersecting point]
PW=yw-------(2).[W is intersecting point]
from 1 and 2,
wx=py,
WS=wy
from 1
wx=wy.
similarly,
xz=zr-----(3) [ z is intersecting point]
qz=yz------(4)[z is intersecting point]
from 3 and 4,
xz=qz
zr=yz
from 3,
xz=yz.
Now in ∆ xwz and ywz,
xw=yw [ from above eq.1]
zx=yz. [ from above eq.3]
wz=wz. [ common]
By sss congruency,
∆ xwz ~= ∆ ywz.
so,
ar(∆xwz)=ar(∆ywz).