Why do the midpoints of a rhombus form a rectangle? Solved in a paragraph proof.

2 Answers
Mar 18, 2018

see explanation.

Explanation:

enter image source here
Some of the properties of a rhombus :
1) all sides are congruent, => AB=BC=CD=DAAB=BC=CD=DA,
2) opposite angles are congruent, => angleADC=angleABC=yADC=ABC=y,
and angleBAD=angleBCD=xBAD=BCD=x,
3) adjacent angles are supplementary, => x+y=180^@x+y=180
4) opposite sides are parallel, => ADAD // BC, and ABBC,andAB // DCDC,

given that P,Q, R and SP,Q,RandS are midpoints of AB,BC,CD, and DAAB,BC,CD,andDA, respectively,
=> AP=PB=BQ=QC=CR=RD=DS=SAAP=PB=BQ=QC=CR=RD=DS=SA
Consider DeltaAPS, as AP=AS, => DeltaAPS is isosceles,
=> angleASP=angleAPS=w,
=> x+2w=180^@ ----- Eq(1)
Consider DeltaBPQ, as BP=BQ, => DeltaBPQ is isosceles,
=> angleBPQ=angleBQS=z
=> y+2z=180^@ ----- Eq(2)
Eq(1)+Eq(2) = (x+y)+2(w+z)=360^@
=> 180+2(w+z)=360^@
=> w+z=90^@
=> angleSPQ=180-(w+z)=180-90=90^@
Similarly, anglePQR=angleQRS=angleRSP=180-(w+z)=90^@

Hence, PQRS is a rectangle.

Mar 18, 2018

enter image source here
Given
In the above figure ABCD is a rhombus where sides AB=BC=CD=DA.
P,Q,R,S are the midpoints of sides AB,BC,CD,DA respectively,

P,Q,R,S are joined in order to form the quadrilateral PQRS .

Rtp
We are to prove PQRS is always rectangle.

Construction
Diagonals AC and BD of the rhombus are drawn.
Proof
Now by midpoint theorem of a triangle any two opposite sides of the rectangle PQRS are parallel and half of the diagonal.

So PQ=SR=1/2AC and PQ||AC, SR||AC
Hence in PQRS, PQ=SR and PQ||SR. Hence PQRS is a parallelogram.

So PS=RQ and PS||RQ||BD
And PQ=SR and PQ||SR||AC

But AC_|_BD as these are diagonals of a rhombus.
Hence adjacent sides of the quadrilateral,which are parallel to diagonals, must be perpendicular to each other, This proves that PQRS is a rectangle.