Question

199. In Fig 5. 50 2 QPR (PQR and Manon.
are respectively points on sides Q Rand
PR of APQR such that QM-PN. N.
Фиои Оло+ OP = 08 , Mahin oks
the point of intensection of
PM and QN
Fig. 5.50
​

Answer 1

here is your answer

Given: ∠QPR = ∠PQR and M and N are respectively points on side QR and PR of ∆PQR, such that QM = PN.

To Prove: OP = OQ, where O is the point of intersection of PM and QN.

Proof: In ∆PNQ and ∆QMP,

PN = QM    | Given

PQ = QP    | Common

∠QPN = ∠PQM    | Given

∴ ∆PNQ ≅ ∆QMP

| SAS congruence rule

∴ ∠PNQ = ∠QMP    | CPCT

Again, in ∆PNO and ∆QMO,

PN = QM    | Given

∠PON = ∠QOM

| Vertically opposite angles

∠PNO = ∠QMO | Proved above

∴ ∆PNO ≅ ∆QMO

| AAS congruence rule

∴ OP = OQ    | CPCT

Hope it help you