Question

ABCD is quadrilateral E,F,G and H are the midpoints of AB,BC,CDandDA respectively .prove that EFGH is a parallelogram.

Answer 1

GIVEN, quad ABCD
E,F,G,H are mid points
To Prove, EFGH is a parallelogram.

Proof,

Join AC and BD,
In ∆ADC

H is mid point of AD and G is mid point of DC.
So by mid point theorem,

HG || AC and HG= AC/2.......(1)

Similarly,

EF ||AC and EF =AC/2........(2)

By (1) & (2),

HG || EF and HG=AF

We know,
If in a quad opposite sides are equal and parallel then it is a parallelogram.

Hence proved

Answer 2

good morning

very nice question and here is your answer

Given :  quad ABCD E,F,G,H are mid points of AB , BC , CD , DA .

To Prove, EFGH is a parallelogram.

Construction : Join daignols AC and BD

In ∆ADC E , H are the midpoints of AD and DC

we know that a line segment joining the mid point of any two sides of a triangle is parallel to the third side and half of it .

HG || AC and HG = AC/2..(1)

Similarly in ΔABC , F AND G are the mid points of AB and BC .

we know that a line segment joining the mid points of any two sides of a triangle is parallel to the third side and half of it .

FE ||AC and FE =AC/2..(2)

From (1) & (2), HG || FE and HG = FE

we know that in any quadrilateral if the opposite sides are equal and parallel it is said to be a parallelogram .

Hence proved that EFGH is a parallelogram

hope this helps you

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