Question

If ABCD is a quadrilateral and E, F, G, H are the midpoints of AB, BC, CD and DA respectively then EFGH is a:

• A. rectangle
• B. square
• C. rhombus
• D. parallelogram

Answer 1

Answer: (D)

parallelogram

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 $△ACD$

$\therefore$ 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 $△ABC$

$\therefore$ 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 $△DCB.$

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

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

$\therefore$ 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.