In the given figure, is a quadrilateral whose diagonals intersect at right angles. Show that the quadrilateral formed by joining the midpoints of the pairs of adjacent sides is a rectangle.

Here, in ABCD, diagonals intersect at 90°


Also, in ABCD, P, Q, R and S be the midpoints of AB, BC, CD and DA, respectively.


In ∆ ABC, we have,
PQ ∣∣ AC and PQ = AC …by midpoint theorem


Similarly, in ∆DAC,


SR ∣∣ AC and SR = AC …by midpoint theorem


Now, PQ ∣∣ AC and SR ∣∣ AC


PQ ∣∣ SR


Also, PQ = SR = AC


Hence, PQRS is parallelogram.


We know that the diagonals of the given quadrilateral bisect each other at right angles.
EOF = 90°
Also, RQ ∣∣ DB
RE ∣∣ FO
Also, SR
∣∣ AC
FR ∣∣ OE
OERF is a parallelogram.


So, FRE = EOF = 90° …Opposite angles of parallelogram are equal
Thus, PQRS is a parallelogram with
R = 90o.
PQRS is a rectangle.


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