In the given figure P and Q are two points which lie on the same side of line XY - Maths - Quadrilaterals

Question

In the given figure P and Q are two points which lie on the same
side of line XY PS perpendicular to XY , QR perpeendicular to XY
and T is the mid point of line segment PQ prove that TS=TR

Manbar Singh · Accepted Answer

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Given PS and QR are perpendiculars to XY T is the midpoint of
PQ
To prove TR=TS
Construction Join PR and draw TL perpendicular to XY

PS⊥XY,TL⊥XY,QR⊥XY∴PS∥TL∥QRNow in ∆PQR,TM∥QR and T is the midpoint of PQ
Hence M is the midpoint of PR (by converse of midpoint theorem)In ∆PRS,M is the midpoint of PR and ML∥PSHence L is the midpoint of RS (by converse of midpoint theorem)In ∆TRL  and ∆TSLRL = SL    as, L is the midpoint of RS∠TLR=∠TLS  90° eachTL=TL     (common)Hence by SAS congruency , ∆TRL  ≅ ∆TSL∴ TR = TS   CPCT

In the given figure P and Q are two points which lie on the same side of line XY. PS perpendicular to XY , QR perpeendicular to XY and T is the mid point of line segment PQ. prove that TS=TR.