Question

If the quadrilateral formed by joining the mid-points of the adjacent sides of quadrilateral $ABCD$ is a rectangle, show that the diagonals $AC$ and $BD$ intersect at right angle. State true or false.

• A. True
• B. False

Answer 1

The correct option is A True

Given: $ABCD$ is a quadrilateral. $P,Q,R$ and $S$ are the mid points of $AB,BC,CD,DA.PQRS$ is a rectangle.
Construction: Join $AC$ and $BD$
Now in $△ACD$
$R$ and $S$ are mid points of $AC$ and $AD$ respectively.
hence, by mid point theorem, $RS\parallel AC$
Now in $△ABC$
$P$ and $Q$ are mid points of $AB$ and $BC$ respectively.
Hence, by mid point theorem, $PQ\parallel AC$
Similarly, $QR\parallel SP\parallel BD$
We know, $PQRS$ is a rectangle, $PQ\perp QR$
Thus, $AC\perp BD$ (Angle made between two lines is same as the angle between their corresponding parallel sides)