Question

ABCD is a quadrilateral in which AC I BD, prove that the quadrilateral formed by joining

the mid-points of consecutive sides of the quadrilateral ABCD is a rectangle.​

Answer 1

Answer:

Given: ABCD is a quadrilateral. P, Q, R and S are the mid points of AB, BC, CD, DA. PQRS is a rectangle.

Construction: Join AC and BD

Now in △ACD

R and S are mid points of AC and AD respectively.

hence, by mid point theorem, RS∥AC

Now in △ABC

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

Hence, by mid point theorem, PQ∥AC

Similarly, QR∥SP∥BD

We know, PQRS is a rectangle, PQ⊥QR

Thus, AC⊥BD (Angle made between two lines is same as the angle between their corresponding parallel sides)

Answer 2

Given : ABCD is a quadrilateral in which AC ⊥ BD,

quadrilateral formed by joining  the mid-points of consecutive sides of the quadrilateral ABCD

To Find : prove that Quadrilateral formed   is a rectangle​

Solution:

ABCD is a quadrilateral in which AC ⊥ BD,

Let say P , Q , R & S are mid points on AB , BC , CD and AD

line joining the mid-point of two sides of a triangle is parallel to third side and equal to half the length of the third side

PQ || AC   and PQ = AC/2

Similarly RS || AC and RS = AC/2

=> PQ = RS   and PQ || RS || AC

and QR || BD  , QR = BD/2

PS || BD ,  PS = BD/2

=> QR = PS   ad QR || PS || BD

AC ⊥ BD

=> PQ ⊥  QR  and PR

     RS ⊥ QR  and PS

Opposite sides are equal and adjacent sides are perpendicular

Hence PQRS is a rectangle.

QED

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