Question

The diagonals of a quadrilateral intersect at
right angles. Prove that the figure obtained by
joining the mid-points of the adjacent sides of
the quadrilateral is a rectangle.​

Answer 1

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Answer:

Here, ABCD is a quadrilateral. AC and BD are diagonals, which are perpendicular to each other.

P,Q, R and S are the mid-point of AB,BC,CD and AD respectively.

In △ABC,

Pand Q are mid points of AB and BC respectively.

∴ PQ∥AC and PQ=1/2AC. - ( 1 ) [ By mid-point theorem ]

Similarly, in △ACD,

R and Sare mid-points of sides CD and AD respectively.

∴SR∥AC and SR=1/2AC ----- ( 2 ) [ By mid-point theorem ]

From ( 1 ) and ( 2 ), we get

PQ∥SR and PQ=SR

∴PQRS is a parallelogram.

Now, RS∥AC and QR∥BD.

⇒ Also, AC⊥BD [ Given ]

∴ RS⊥QR

∴ PQRS is a rectangle.

Answer 2

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