Question

If the vertices of a quadrilateral $ABCD$ are $A(-4,2),B(2,6),C(8,5)$ and $D(9,-7)$, then the quadrilateral formed by joining the midpoints of the sides of the quadrilateral $ABCD$ will be a

• A. Rhombus
• B. Parallelogram
• C. Rectangle
• D. Square

Answer 1

The correct option is B Parallelogram

The vertices of quadrilateral are
$A(-4,2),B\left(2.6\right),C(8,5),D(9,-7)$

Mid points of $AB,BC,CD$ and $DA$ are
$P(-1,4),Q(5,\frac{11}{2}),R(\frac{17}{2},-1),S(\frac{5}{2},-\frac{5}{2})$
$m_1=$ Slope of $PQ=\frac{1}{4}$
$m_2=$ Slope of $RS=\frac{1}{4}$
$m_3=$ Slope of $QR=-\frac{13}{7}$
$m_4=$ Slope of $PS=-\frac{13}{7}$
Clearly $m_1=m_2$ and $m_3=m_4$ $m_1{m}_3\ne -1$ and
$PQ=RS=\frac{\sqrt{153}}{2},QR=PS=\frac{\sqrt{218}}{2}$
So, $PQ\ne QR$
So, it is a parallelogram