We get a rhombus by joining the mid-points of the sides of a ____.

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We get a rhombus by joining the mid-points of the sides of a rectangle.Let ABCD is a rectangle such as AB = CD and BC = DA. P, Q, R and S are the mid points of the sides AB, BC, CD and DA respectively.Join AC and BD.In

ΔABC
 , P and Q are the mid-points of AB and BC respectively.Midpoint theorem states that the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and equal to half of the length of the third side.Therefore,

PQ||AC
  and

PQ=
    1
    2
   
   AC
 .                    ……(1)Similarly, in

ΔADC
 ,

SR||AC
  and

SR=
    1
    2
   
   AC
 .    ……(2)Clearly from (1) and (2),

PQ||SR
  and PQ = SR.Since, in quadrilateral PQRS one pair of opposite sides is equal and parallel to each other, it is a parallelogram.Since, opposite sides of a parallelogram are equal, therefore, PS || QR and PS = QR.                                             ……(3)In ΔBCD, Q and R are the mid-points of side BC and CD respectively.By mid-point theorem,

QR||BD,so QR=
    1
    2
   
   BD
 .                                           ……(4)However, the diagonals of the rectangle are equal.So,

AC=BD
 .                                                           ……(5)By using (1), (2), (3), (4) and (5), we get,PQ = QR = SR = PS.So, PQRS is a rhombus.Therefore, from the rectangle, rhombus is formed by joining the midpoints.

We get a rhombus by joining the mid-points of the sides of a ____.

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