Question -
Answer -
Given:
The diagonal AC and BDof the quadrilateral ABCD, intersect each other at point E.
Construction:
From A, draw AMperpendicular to BD
From C, draw CNperpendicular to BD
To Prove,
ar(ΔAED) ar(ΔBEC) = ar(ΔABE) ×ar (ΔCDE)
Proof,
ar(ΔABE) = ½×BE×AM………….. (i)
ar(ΔAED) = ½×DE×AM………….. (ii)
Dividing eq. ii by i ,we get,
ar(AED)/ar(ABE) =DE/BE…….. (iii)
Similarly,
ar(CDE)/ar(BEC) =DE/BE ……. (iv)
From eq. (iii) and(iv) , we get
ar(AED)/ar(ABE) =ar(CDE)/ar(BEC)
, ar(ΔAED)×ar(ΔBEC) =ar(ΔABE)×ar (ΔCDE)
Hence proved.