The figure formed by joining the mid-points of the adjacent sides of a rectangle is a ____.

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Detailed Solution

The figure formed by joining the mid-points of the adjacent sides of a rectangle is a rhombus.
Construct a rhombus

A B C D

To prove, PQRS is a rectangle, it is sufficient to show that it is a parallelogram, whose one angle is a right angle.
In ΔABC, P and Q are the mid-points of AB and BC respectively.

P Q | | A C a n d P Q = 12 A C

……(1)
In ΔADC, R and S are the mid-points of CD and AD respectively.

R S | | A C a n d R S = 12 A C

……(2)
From (1), (2),

P Q ∥ R S, P Q = R S

……(3)
So, PQRS is a parallelogram.
Now, ABCD is a rectangle.
So,

A D = B C ⇒ 12 A D = 12 B C ⇒ A S = B Q

Consider

Δ A P S, Δ B P Q

∠ P A S = ∠ P B Q

[right angle]
Since P is the midpoint of AB, PA =PB.
And AS = BQ.
By SAS congruence theorem,

Δ A P S ≅ Δ B P Q

By corresponding parts of congruency triangles,
PS = PQ
Therefore, the adjacent sides are equal.
So, PQRS is a rhombus.
Thus the required answer is rhombus.

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