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.

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.
PSXY,TLXY,QRXYPSTLQRNow in PQR,TMQR 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 MLPSHence L is the midpoint of RS (by converse of midpoint theorem)In TRL  and TSLRL = SL    as, L is the midpoint of RSTLR=TLS  90° eachTL=TL     (common)Hence by SAS congruency , TRL   TSL TR = TS   CPCT

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