Question

P,q,r,s are respectively the mid points of sides ab,bc,cd,da of a quadrilateral abcd, such that ac is perpendicular to bd. prove that pqrs is a rectangle.

Answer 1

Given :

• A quadrilateral ABCD in which P, Q, R, S are the mid points of sides AB , BC , CD , DA respectively .

• AC is the diagonal.

To prove :

• SR//AC and SR= ½AC
• PQ= SR
• PQRS is a parallelogram .

Proof :

In ∆ACD , S is the mid point of AD and R is the mid point of DC .

# Hence SR//AC and SR= ½AC -(1)

: By Mid point Theorem .


Now , in ∆ ABC , P is the mid point of AB and Q is the mid point of BC .

# Hence PQ//AC , PQ =½AC -(2)

By (1)&(2) ,

SR = PQ , SR //PQ -(3)

By (3) we can say that PQRS is a parallelogram .

#Hence Proved .

BE BRAINLY!!

Answer 2

here is your answer OK dude

☺☺☺☺☺

Since , ABCD is a quad. and R,S are the midpoints ,then

RS is parallel to BD and

similarly, QR is parallel to BD

PQ is parallel to RS and

PS is parallel to QR.

that is PQRS is a parallelogram.

Now, AC is perp. to BD.

therefore, ∠1=90°,∠2=90°,∠3=90°,∠4=90°

Hence, PQRS is a rectangle.

OK I hope I help you