Question

PQR is an isosceles triangle in which PQ=PR.Side QP is produced to S such that PS=PQ. show that QRS is right triangle ​

Answer 1

Answer:

In trig PQR

PQ =PR

Step-by-step explanation:

In Trig PQS & trig PRS

PQ =PR ( given)

PS =PS ( common)

< PSQ = < PSR (90°)

trig PSQ congruence trig PRS {RHS}

QS=RS

PS bisect QR

Answer 2

Answer:

Proved QRS is right triangle

Step-by-step explanation:

Concept= Isosceles and Angle

Given= Isosceles triangle, Side equality

To find= Prove a right triangle

Explanation=

We have been given that

PQR is an isosceles triangle in which PQ=PR.

Side QP is produced to S such that PS=PQ.

Now, since in ΔPQR it is isosceles and PQ=PR therefore

∠PQR = ∠PRQ= x(let)

So according to sum of angle of triangle PQR which is 180° we have

∠PQR + ∠PRQ + ∠QPR =180

x+x+ ∠QPR =180

∠QPR = 180 - 2x.

now if we see the line QS the sum of ∠QPR and ∠RPS will be 180°

∠QPR + ∠RPS=180

180-2x +∠RPS= 180

∠RPS= 2x.

In ΔRPS, PS=PQ and since PQ=PR so PR=PS. hence ΔRPS is isosceles.

∠PRS = ∠PSR and ∠RPS + ∠PRS + ∠PSR= 180°

∠PRS + ∠PSR +2x=180

∠PRS + ∠PSR= 180-2x

∠PRS + ∠PRS =180-2x

∠PRS= 90-x.

Now coming to ΔQRS

∠PRQ + ∠PRS= ∠QRS

x + 90-x=∠QRS

90°=∠QRS.

Since the angle ∠QRS in ΔQRS is 90° the triangle is right angled.

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