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

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Detailed Solution

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.
Question Image

Δ A B C

, 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,

P Q | | A C

and

P Q = 12 A C

. ……(1)
Similarly, in

Δ A D C

S R | | A C

and

S R = 12 A C

. ……(2)
Clearly from (1) and (2),

P Q | | S R

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,

Q R | | B D, s o QR= 12 B D

. ……(4)
However, the diagonals of the rectangle are equal.
So,

A C = B D

. ……(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.

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