Q. 174.4( 13 Votes )

In Fig., PQRS<

Answer :

Given,


PQRS is a square and SRT is a equilateral triangle


To prove: (i) PT = QT


(ii) TQR =15 �


Proof: PQ = QR = RS = SP (Given) (i)


And, SPQ = PQR = QRS = RSP = 90o


And,


SRT is an equilateral triangle


SR = RT = TS (ii)


And, TSR = SRT = RTS = 60o


From (i) and (ii)


PQ = QR = SP = SR= RT = TS (iii)


TSP = TSR + RSP


= 60o + 90o = 150o


TRQ = TRS + SRQ


= 60o + 90o = 150o


Therefore, TSR = TRQ = 150o (iv)


Now, in ∆TSP and ∆TRQ, we have


TS = TR (From iii)


TSP = TRQ (From iv)


SP = RQ (From iii)


Therefore, By SAS theorem,


∆TSP TRQ


PT = QT (BY c.p.c.t)


In ∆TQR


QR = TR (From iii)


Hence, ∆TQR is an isosceles triangle.


Therefore, QTR = TQR (Angles opposite to equal sides)


Now,


Sum of angles in a triangle is 180o.


QTR + TQR + TRQ = 180O


2TQR + 150O = 180O (From iv)


2TQR = 30O


TQR = 15O


Hence, proved


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