Question

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Answer 1

Question:-

In the given figure, ZQPR = ZPQR and M and N are respectively points on the sides QR and PR of APQR, such that QM = PN. Prove that OP=OQ, where O is the point of intersection of PM and QN.

Answer:-

Given:- In ∆PQR, ∠QPR = ∠PQR, M and N are two pointś on OR and PR such that QM = PN, PM and QN intersect at O.

To Prove:- OP = OQ

Proof :- ∆PQM and ∆PQN .

we Have,

$\texttt{QM = PN}$

$∴∠QPM = ∠QPN \: \text{[∠QPR = ∠QPR}$

$\text{PQ = PQ [common]}$

$∴∆PQM \cong \: PQN \: \text{[SAS congruence rule]}$

$∠PQM =∠ PQN$

$\text{But ∠PQN = ∠PQM}$

$⇒\text{∠QPN - ∠QOM = ∠PQM - ∠PQM}$

$⇒\text{∠OPN = ∠OQM}$

$\text{Again in ∆PON and ∆QOM }$

$\text{PN = QM [Given]}$

$∴\text{∠OPN = ∠OQM [As Proved]</p><p>}$

$\text{∠PON = ∠QOM [Vertically opposite angles]}$

$∴∆PON\cong ∆QOM \: \text{[AAS congruence Rule]}$

$\textbf{OP = OQ}$

$\textbf{Hence Proved}$

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