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Question

State true or false:
If the diagonals of a quadrilateral intersect at right angles then the figure obtained by joining the mid-points of the adjacent sides of the quadrilateral is a rhombus.

A
True
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B
False
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Solution

The correct option is B False
Given: ABCD is a quadrilateral. AC and BD intersect at right angles. P, Q, R, S are mid points of respective sides.
In ABC,
PQAC and PQ=12AC (By mid point theorem)
In ADC,
SRAC, and SR=12AC (By mid point theorem)
Therefore, PQ=SR and PQSR
Similarly, using diagonal DC, PS=RQ and PSRQ
hence, PQRS is a parallelogram.
Since, PQAC and QRBD
But, ACBD
Hence, PQQR (Angles between two lines = Angles between their parallels)
Thus, PQRS is a rectangle.
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