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

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Detailed Solution

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

A B C D

Join A and C.
Let P and Q be the midpoints of sides AB and BC respectively.
In

Δ A B C

P Q ∥ A C

.
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.
So, by the above theorem,

P Q = 12 A C

……(1)
S and R are the midpoints of AD and DC respectively.
In

Δ A C D

S R ∥ A C

.
By midpoint theorem,

S R = 12 A C

……(2)
From (1) and (2),

P Q ∥ S R

and PQ = SR.
Since, one pair of opposite sides are parallel and equal.
So, PQRS is a parallelogram.

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