Question

In the figure P,Q,R,Sis a square and S,R,T is an Quadrilateraltriangle and prove that PT =QT

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Answer 1

${\underline{\underline{\large{\sf{\pink{\pink{\bigstar{ \: Given:}}}}}}}}$

• PQRS is square.
• SRT is equilateral triangle.

${\underline{\underline{\large{\sf{\pink{\pink{\bigstar{ \: To \: prove :}}}}}}}}$

• PT = QT

${\underline{\underline{\Large{\sf{\pink{\red{\bigstar{ \: Solution :}}}}}}}}$

According to the question we know that all sides of square is Equal and parallel to each other and all angle are 90°. And also in equilateral triangle all sides are equal and angles are 60°.

As PQRS is a square, so,

⟹ ∠PSR=∠QRS ..............(each 90° )

Also, ΔSRT is an equilateral triangle, then,

⟹ ∠TSR=∠TRS ................(each 60° )

Now,

⟹ ∠PSR+∠TSR=∠QRS+∠TRS

⟹ ∠TSP=∠TRQ

In ΔTSP and ΔTRQ,

》TS=TR .............. (sides of equilateral triangle)

》∠TSP=∠TRQ

》PS=QR ............... (sides of square)

So, by SAS congruence rule,

ΔTSP ≅ ΔTRQ

By CPCT means Congruence part of congruent triangle.

》 PT=QT

Hence, proved.

Answer 2

Given:

PQRS is a square and SRT is an equilateral triangle.

To Prove :

PT=QT

$\huge\bold{\gray{\sf{Answer:}}}$

Explanation:

➠∠QRT=90°+60°=150°

Reason :90° [PQRS is a square and in square angle made at every side is 90°]

60°[SRT is an equilateral triangle and we know that equilateral triangle having all 3 sides equal and angles are also equal which is 60° each ]

➠∠PST=90°+60°=150°

➠In ∆QRT and ∆PST

➠PS=QR (side of square)

➠ST=RT(sides of an equilateral triangle)

$\bold{ ∴∠QRT=PST}$$\bold{\red{ [By SAS Rule ]}}$

so,∆QRT≅ ∆PST

∴QT=PT(CPCT)