Question

In figure, PQRS and MNRL are rectangles. If point M is the midpoint of side PR then prove that, $\left(i\right)SL=LR,\left(ii\right)LN=\frac{1}{2}SQ.$

Answer 1

(i) Given PQRS and MNRL are rectangles.
$S={90}^{\circ }\text{[Angle of rectangle]}$
$L={90}^{\circ }\text{[Angle of rectangle]}$
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)]
$LN=\frac{1}{2}SQ\text{[Midpoint theorem]}$
Hence proved.