Question

ABCD is a square. P, Q and R are the points on
AB, BC and CD respectively; such that
AP = BQ = CR. Prove that :
(i) PB = QC
(iii) If angle PQR is a rt. angle; find angle
PRQ.
(ii) PQ = QR​

Answer 1

Answer:

Given: ABCD is a square.

thus, AB=BC

AP=BQ (Given)

AB−AP=BC−BQ

PB=CQ (I)

Now, In △PBQ and △CQR,

∠PBQ=∠QCR (Each 90

∘

)

PB=CQ (From I)

BP=CR (Given)

thus, △PBQ≅△CQR (SAS rule)

or, PQ=QR (By cpct)

Now, In △PQR

PQ=QR

thus, ∠PRQ=∠QPR=x (Isosceles triangle property)

∠PQR=90

∘

(Given)

Sum of angles = 180

∠PQR+∠QPR+∠QRP=180

90+x+x=180

2x=90

x=45

Thus, ∠PRQ=45 degree

Hope this will help you ❤️

Answer 2

Step-by-step explanation:

Given: ABCD is a square.

thus, AB=BC

AP=BQ (Given)

AB−AP=BC−BQ

PB=CQ (I)

Now, In △PBQ and △CQR,

∠PBQ=∠QCR (Each 90

∘

)

PB=CQ (From I)

BP=CR (Given)

thus, △PBQ≅△CQR (SAS rule)

or, PQ=QR (By cpct)

Now, In △PQR

PQ=QR

thus, ∠PRQ=∠QPR=x (Isosceles triangle property)

∠PQR=90

∘

(Given)

Sum of angles = 180

∠PQR+∠QPR+∠QRP=180

90+x+x=180

2x=90

x=45

Thus, ∠PRQ=45

∘