The diagonals of a quadrilateral ABCD are perpendicular. Show that the quadrilateral, formed by joining the mid-points of its sides, is a rectangle.
Given: A quadrilateral whose diagonals AC and BD are perpendicular to each other. P, Q, R, S are the mid-points of sides. AB, BC, CD and DA respectively. PQ, QR, RS and SP are joined.
To Prove: PQRS is a rectangle.
Proof: In ΔABC, P and Q are the mid-points of AB and BC respectively.
... PQ ïï AC and PQ =
AC........(i)
In the ΔADC, R and S are the mid-points of CD and AD respectively.
... RS ïï AC and RS =AC........(ii)
From (i) and (ii), we have
PQ ïï RS and PQ = RS
Thus, in quadrilateral PQRS, a pair of opposite sides are equal and parallel. So, PQRS is a parallelogram. Suppose the diagonals AC and BD of quadrilateral ABCD intersect at O. Now in ΔABD, P is the mid-points of AB and S is the mid-point of AD.
... PS ïï BD ⇒ PN ïï MO
Also, from (i), PQ ïï AC ⇒ PM ïï NO
Thus in quadrilateral PMON,
we have
PN ïï MO and PM ïï NO ⇒ PMON is a parallelogram ⇒ ∠MPN = ∠MON [opp. angles of parallelogram are equal]
⇒ ∠MPN = ∠BOA [... ∠BOA = ∠MON] ⇒ ∠MPN = 90° [...AC ⊥ BD, ... ∠BOA = 90°]
⇒ ∠QPS = 90° [...∠MPN = ∠QPS]
Thus, PQRS is a parallelogram
whose one angle ∠QPS = 90°.
Hence, PQRS is a rectangle.
To Prove: PQRS is a rectangle.
Proof: In ΔABC, P and Q are the mid-points of AB and BC respectively.
... PQ ïï AC and PQ =
AC........(i)In the ΔADC, R and S are the mid-points of CD and AD respectively.
... RS ïï AC and RS =
AC........(ii)From (i) and (ii), we have
PQ ïï RS and PQ = RS
Thus, in quadrilateral PQRS, a pair of opposite sides are equal and parallel. So, PQRS is a parallelogram. Suppose the diagonals AC and BD of quadrilateral ABCD intersect at O. Now in ΔABD, P is the mid-points of AB and S is the mid-point of AD.
... PS ïï BD ⇒ PN ïï MO
Also, from (i), PQ ïï AC ⇒ PM ïï NO
Thus in quadrilateral PMON,
we have
PN ïï MO and PM ïï NO ⇒ PMON is a parallelogram ⇒ ∠MPN = ∠MON [opp. angles of parallelogram are equal]
⇒ ∠MPN = ∠BOA [... ∠BOA = ∠MON] ⇒ ∠MPN = 90° [...AC ⊥ BD, ... ∠BOA = 90°]
⇒ ∠QPS = 90° [...∠MPN = ∠QPS]
Thus, PQRS is a parallelogram
whose one angle ∠QPS = 90°.
Hence, PQRS is a rectangle.
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