## In the given figure, PQR is a triangle in which PQ = PR. In the given figure, PQR is a triangle in which PQ = PR. QM and RN are the QM and R

Question

In the given figure, PQR is a triangle in which PQ = PR. In the given figure, PQR is a triangle in which PQ = PR. QM and RN are the QM and RN are the medians of the triangle. Prove that medians of the triangle. Prove that (i) ΔNQR ≅ ΔMRQ (ii) QM = RN please

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2 weeks 2021-09-10T21:31:06+00:00 2 Answers 0 views 0

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Solution:

ΔPQR is an isosceles triangle. [∵ PQ = PR]

⇒ 1/2 PQ = 1/2 PR

⇒ NQ = MR and PN = PM

(i) In ΔNQR and ΔMRQ

NQ = MR (Half of equal sides)

∠NQR = ∠MRQ (Angles opposite to equal sides)

QR = RQ (Common)

ΔNQR = ΔMRQ (By SAS rule)

(ii) QM = RN (Congruent parts of congruent triangles)

(iii) In ΔPMQ and ΔPNR

PN = PM (Half of equal sides)

PR = PQ (Given)

∠P = ∠P (Common)

ΔPMQ = ΔPNR (By SAS rule)

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ΔPQR is an isosceles triangle. [∵ PQ = PR]

⇒ 1/2 PQ = 1/2 PR

⇒ NQ = MR and PN = PM

(i) In ΔNQR and ΔMRQ

NQ = MR (Half of equal sides)

∠NQR = ∠MRQ (Angles opposite to equal sides)

QR = RQ (Common)

ΔNQR = ΔMRQ (By SAS rule)

(ii) QM = RN (Congruent parts of congruent triangles)

(iii) In ΔPMQ and ΔPNR

PN = PM (Half of equal sides)

PR = PQ (Given)

∠P = ∠P (Common)

ΔPMQ = ΔPNR (By SAS rule)