Question

In the given figure, $\square$ PQRS and $\square$MNRL are rectangles. If point M is the midpoint of side PR then prove that,

(i) SL = LR, (ii) LN = $\frac{1}{2}$SQ.

Answer 1

(i) Given that M is the midpoint of PR. .....(1)
$\angle$PSR = $\angle$MLR = 90$°$ (Since $\square$ PQRS and $\square$MNRL are rectangles)
Thus, LM || SP (Corresponding angles are equal) .....(2)
From (1) and (2) we have
L is the midpoint of SR (Converse of midpoint theorem)
Thus, SL = LR

(ii)
$\angle$RNM = $\angle$RQP = 90$°$ (Since $\square$ PQRS and $\square$MNRL are rectangles)
Thus, MN || PQ (Corresponding angles are equal) .....(4)
From (1) and (4) we have
N is the midpoint of RQ (Converse of midpoint theorem)
Join LN and SQ

By midpoint theorem,
LN || SQ and LN = $\frac{1}{2}$SQ