Usha drew a quadrilateral EFGH in such a way that EF = EH and GF = GH.

Which of these is DEFINITELY true about its diagonals EG and FH?1 they are equal in length2 they bisect each other3 they are perpendicular to each other4 (We cannot say any of these.)

## Given: $ABCD$ is a quadrilateral.

Points $E,F,G,H$ are the midpoints of $AB,BC,CD$ and $DA.$

Draw diagonals $AC$ and $BD$ in the quadrilateral $ABCD$

Segment $HG$ is the midpoint segment in $△ACD$

$∴$ segment $HG$ is parallel to the side $AC$ of the

$△ACD$ (Line segment joining midpoints of two sides of a triangle property.)

Similarly, Segment $EF$ is the midpoint segment in $△ABC$

$∴$ Segments $EF$ is parallel to side $AC$ of $△ABC.$

Since, Segment $HG$ and $EF$ are both parallel to the diagonal $AC$, they are parallel to each other.

Segment $GF$ is the midpoint segment in $△DCB.$

$∴$ Segment $GF$ is parallel to side $DB$ of $△ABD.$

Segment $HE$ is midpoint segment in $△ABD$

$∴$ Segment $HE$ is parallel to side $DB$ of triangle $ABD$

Since,Segment $GF$ and $HE$ are both parallel to diagonal $DB$, they are parallel to each other.

Thus, we have proved that in quadrilateral $EFGH$ the opposite sides

$HG$ and $EF$,$HE$ and $GF$ are parallel by pairs.

Hence, the quadrilateral $EFGH$ is the parallelogram.

So, The answer is they both bisect each other