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PQRS is a square. N and M are midpoints of sides SR and QR respectively. O is a point on diagonal PR such that OP=OR. Show that ONRM is a square also find the ratio of ar(ΔORM) and ar(PQRS).

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by dhruvivaja3

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by dhruvivaja3

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In ΔSRQ,O and M are mid point of side SQ and QR

by mid point theorm

0M// SR and OM=1/2SR=NR

similarly

IN ΔSQR,by mid point theorm

ON//RQ and ON=1/2RS=RM

apposite sides are equal and parallel hence NOMR is a square

In ΔSOR

ΔSON≡ΔNOR

SN=NR

ON=ON

angle 90 degree in both

similarlyΔSON≡NOR≡ROM≡MOQ

4area of ΔSON=1/2 area of PQRS

area of ΔSON=1/8 area of PQRS

area of ΔORM/area of sqPQRS=1/8