Question

PQRS is a square and SRT is an equilateral triangle.prove that PT=QT

Answer 1

PQRSis square hence

PQ=QR=RS=SP ........(1)

∆SRT is equilateral triangle

therefore RS=RT=TS .......(2)

P_T_Q

Two triangles are formed

∆PTS & ∆QTR

IN ∆PTS & ∆QTR

angle SPT=angle RQT .........[each measure 90 degree]

side PS=SideQR........[ From (1)]

segment ST=segment RT.....[from(2)]

therefore ∆PTS congruent to ∆QTR

..............(by hypotenuse side test of congruent triangles)

therefore seg PT= SEG QT....(C.S.C.T)

PT=QT

Answer 2

$\textbf{\huge{ANSWER:}}$

$\sf{Given:}$

PQRS is a square

Triangle SRT is an equilateral

$\sf{To\:Prove:}$

PT = QT

$\sf{Proof:}$

●Since PQRS is a square ( Given ), we can say that :-

PQ = QR = RS = SP ( sides of a square are all equal ) ....(1)

●Since Triangle SRT is equilateral ( Given ), we can say that :-

SR = RT = ST ( sides of an equilateral triangle are all equal ) ....(2)

■Compare (1) and (2). We get that :-

PQ = QR = SP = SR = RT = ST

●Since Triangle SRT is equilateral, we can say that :-

/_T = /_TSR = /_TRS = 60° ( Angles of an equilateral triangle are all equal to 60° )

●Since PQRS is a square, we can say that :-

/_P = /_Q = /_R = /_S = 90° ( Angles of a square are all equal to 90° )

■/_PST = /_S - /_TSR = 90° - 60° = 30°

Similarly, /_QRT = 30°

⊙/_PST = /_QRT

In Triangle PTS and Triangle QRT :-

PS = QR ( Proved )

ST = RT ( Proved )

/_PST = /_QRT ( Proved )

Triangle PTS Is congruent to Triangle QRT by SAS congruence criteria.

Hence Proved!