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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