Question

If the diagonals of a quadrilateral intersect at right angles then the figure obtained by joining the mid-points of the adjacent sides of the quadrilateral is a rectangle.
State whether the above statement is true or false.

• A. True
• B. False

Answer 1

The correct option is A True

Given: ABCD is a quadrilateral. AC and BD intersect at right angles. P, Q, R, S are mid points of respective sides.
In $△ABC$,
$PQ\parallel AC$ and $PQ=\frac{1}{2}AC$ (By mid point theorem)
In $△ADC$,
$SR\parallel AC$, and $SR=\frac{1}{2}AC$ (By mid point theorem)
Therefore, $PQ=SR$ and $PQ\parallel SR$
Similarly, $PS=RQ$ and $PS\parallel RQ$
hence, PQRS is a parallelogram.
Since, $PQ\parallel AC$ and $QR\parallel BD$
But, $AC\perp BD$
Hence, $PQ\perp QR$ (Angles between two lines = Angles between their parallels)
Thus, $PQRS$ is a rectangle.