Question

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

i. SL = LR 

ii. LN = (1/2) SQ

Answer 1

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]