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.

Let ABCD be a quadrilateral and P, Q, R and S are the mid-point of AB, BC, CD and DA.

Diagonal AC and BD intersect at point O.

By mid-point theorem,

The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and is equal to half of it.

In △ ABC,

P and Q are mid-points of AB and BC.

∴ PQ || AC (By mid-point theorem)

From figure,

⇒ ∠AOD = ∠PXO (Corresponding angles are equal) …………(1)

In △ BCD,

R and Q are mid-points of CD and BC.

∴ QR || BD (By mid-point theorem)

Interior angles of a rectangle equals to 90°.

⇒ ∠RQX = ∠Q = 90°.

From figure,

⇒ ∠PXO = ∠RQX = 90° (Corresponding angles are equal) …………(2)

From (1) and (2), we get :

⇒ ∠AOD = ∠PXO = 90°.

Hence, AC and BD intersect at right angles.