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, PQ∥AC and PQ=12AC (By mid point theorem) In △ADC, SR∥AC, and SR=12AC (By mid point theorem) Therefore, PQ=SR and PQ∥SR Similarly, using diagonal DC, PS=RQ and PS∥RQ hence, PQRS is a parallelogram. Since, PQ∥AC and QR∥BD But, AC⊥BD Hence, PQ⊥QR (Angles between two lines = Angles between their parallels) Thus, PQRS is a rectangle.