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.
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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