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Question

In figure, PQRS and MNRL are rectangles. If point M is the midpoint of side PR then prove that, (i)SL=LR, (ii)LN=12SQ.

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Solution

(i) Given PQRS and MNRL are rectangles.
S=90 [Angle of rectangle]
L=90[Angle of rectangle]
RN || LM …(i) [Opposite sides of rectangle MNRL]
RN || SP ….(ii) [Opposite sides of rectangle PQRS]
From (i) and (ii)
LM || SP …..(iii)
In PRS,
M is the midpoint of PR. [Given]
LM || SP [From (iii)]
L is the midpoint of SR …(iv) [Converse of midpoint theorem]
SL = LR
Hence proved.
(ii) Similarly for PQR we can prove that
N is the midpoint of QR …(v).
So in RSQ,
N and L are the midpoints of RQ and SR respectively.
[From (iv) and (v)]
LN=12SQ [Midpoint theorem]
Hence proved.

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