Q. 205.0( 4 Votes )

AB || DC of a trapezium ABCD and E is midpoint of BC. Let’s prove that area of triangular region AED = × area of trapezium shaped region ABCD.

Answer :

Given.


AB || DC of a trapezium ABCD and E is midpoint of BC


Formula used.


Median of triangle divides it into 2 equal parts



As E is midpoint of BC


In triangle ABC


AE is median


Triangle ABE = × triangle ABC


2 × triangle ABE = triangle ABC ……eq 1


In triangle BDC


DE is median


Triangle DEC = × triangle DBC


2 × triangle DEC = triangle DBC ……eq 2


In triangle ADC and triangle DBC


Both are on same base DC


And AB || CD


Triangle ADC = triangle DBC


Putting value from eq 2


Triangle ADC = 2 × triangle DEC ……eq 3


Add Eq1 and Eq 3


We get ;


triangle ADC + triangle ABC = 2 × [Δ ABE + triangle DEC]


Trapezium ABCD = 2 × [trapezium ABCD – triangle AED]


2 × triangle AED =trapezium ABCD


Area of triangle AED = × trapezium ABCD


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