Question

the diagonals of a quadrilateral intersect at right angles. prove that the figure obtained by joining the midpoints of the adjacent sides of the quadrilateral is a rectangle

Answer 1

Answer:

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,

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

∴ PQ∥AC and PQ=

2

1

AC ----- ( 1 ) [ By mid-point theorem ]

Similarly, in △ACD,

R and S are mid-points of sides CD and AD respectively.

∴ SR∥AC and SR=

2

1

AC ----- ( 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.