Question

State true or false:

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.

• 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 CD and DA 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)