Given: PQRS and MNRL are rectangles. M is the midpoint of side PR.
To prove:
i. SL = LR
ii. LN = (1/2) (SQ)
Proof:
i. PQRS and MNRL are rectangles. [Given]
∴ ∠S = ∠L = 90° [Angles of rectangles]
∠S and ∠L form a pair of corresponding angles on sides SP and LM when SR is their transversal.
∴ eg ML || seg PS …(i) [Corresponding angles test]
In ∆PRS,
Point M is the midpoint of PR and seg ML || seg PS. [Given] [From (i)]
∴ Point L is the midpoint of seg SR. …… (ii) [Converse of midpoint theorem]
∴ SL = LR
ii. Similarly for ∆PRQ, we can prove that,
Point N is the midpoint of seg QR. …. (iii)
In ∆RSQ, Points L and N are the midpoints of seg SR and seg QR respectively. [From (ii) and (iii)]
∴ LN = (1/2) SQ [Midpoint theorem]