Given a //gm ABCD where X and Y are the mid points of sides BC and CD respectively. Prove that: 

Ar(AXY)  = 3/8× ar. (//gm ABCD) 

Asked by saumya15crazy | 14th Feb, 2018, 12:27: AM

Expert Answer:

begin mathsize 16px style Join space poin increment ts space straight B space and space straight D.
Since space straight X space and space straight Y space are space the space mid space points space of space side space BC space and space CD space respectively.
In space increment BCD comma space XY parallel to BD space rightwards double arrow space XY equals 1 half BD.
Similarly comma
ar left parenthesis increment CYX right parenthesis equals 1 fourth ar left parenthesis increment DBC right parenthesis space and space ar left parenthesis increment CYX right parenthesis equals 1 over 8 ar left parenthesis increment DBC right parenthesis
As space area space of space parallelogram space is space twice space the space area space of space triangle space made space by space diagonal.
parallelogram space ABCD space and space increment ABX space are space between space same space parallel space lines space AD space and space BC space and space BX equals 1 half BC
ar open parentheses increment ABX close parentheses equals 1 fourth ar open parentheses parallelogram space ABCD close parentheses
Also space ar open parentheses increment AYD close parentheses equals 1 fourth ar open parentheses parallelogram space ABCD close parentheses
ar open parentheses increment AXY close parentheses equals ar open parentheses parallelogram space ABCD close parentheses minus left square bracket ar open parentheses increment ABX close parentheses plus ar open parentheses increment AYD close parentheses plus ar left parenthesis increment CYX right parenthesis right square bracket
After space putting space values space of space ar open parentheses increment ABX close parentheses comma ar open parentheses increment AYD close parentheses comma ar left parenthesis increment CYX right parenthesis
solve space it space further space you space get space
ar open parentheses increment AXY close parentheses equals 3 over 8 Parallelogram space ABCD
end style

Answered by Sneha shidid | 14th Feb, 2018, 11:05: AM