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ABCD is a given quadrilateral whose diagonals intersect at right angles. If midpoints of the pair of adjacent sides are joined then the quadrilateral formed is a

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Square

Rectangle

Trapezium

Kite

answer is B.

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Detailed Solution

ABCD is the given quadrilateral and we also know that its diagonals intersect at the right angle.
Question Image

We have to find out what type of quadrilateral PQRS is.
In ABCD, DB and AC are diagonals that intersect at

90 °

.
As given in the figure P, Q, R, and S are midpoints of AB, BC, CD, and DA respectively.
In,

Δ A B C

, PQ||AC.
So,

P Q = 12 A C

... (by midpoint theorem)
Similarly, in

Δ D A C

, SR||AC.
So,

S R = 12 A C

… (by midpoint theorem)
Since, PQ||AC and SR||AC.
Therefore, PQ||SR and,

P Q = S R S R = 12 A C

(From above)
Therefore, PQRS is a parallelogram.
Further,

∠ E O F = 90 °

.
Also, RQ||DB, RE||FO, SR||AC and FR||OE.
Thus, OERF is a parallelogram.
Therefore,

∠ F R E = ∠ E O F ∠ E O F = 90 °

. (Opposite angles of a parallelogram are equal)
So, PQRS is a parallelogram where

∠ R = 90 °

.
This gives us that PQRS is a rectangle.
Hence, option 2 is the correct choice.

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