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Question
In the given figure, `square`PQRS and `square`MNRL are rectangles. If point M is the midpoint of side PR then prove that,
- SL = LR
- LN = `1/2`SQ
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Solution
(i) `square`LMNR and `square`MNRL are rectangles.
∴ Side LM || Side RN ...(Opposite sides of rectangle)
That is, Side LM || Side RQ ...(R-N-Q) ...(i)
Side RQ || Side SP ...(Opposite sides of the rectangle) ...(ii)
From (i) and (ii),
Side LM || Side SP ...(iii)
In ΔRSP,
Point M is the midpoint of Seg PR.
Line LM || Line SP ...[From (iii)]
∴ Point L is the midpoint of Seg SR. ...(Converse of Midpoint Theorem) ...(iv)
∴ SL = LR
(ii) The diagonals of a rectangle are congruent.
∴ SQ = PR ...(v)
LN = MR ...(vi)
Now, MR = `1/2` PR ...(Point M is the midpoint of line PR.) ...(vii)
∴ LN = `1/2` PR ...[From (vi) and (vii)] ...(viii)
∴ LN = `1/2` SQ ...[From (vii) and (viii)]
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