Question

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

Answer 1

Answer:

(i) 1st property is proven by the isoceles triangle or bpt theorem

which is

pq=pr given

rn and qm are medians therefore nq=mr--------------(a)

angle q=angle r

proved by sas criteria

(ii) (a) proved above

(iii) as nq=mr

therefore pn=pm

also angle n =angle m

by sas criteria this also proved

Step-by-step explanation:

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Answer 2

Given:

PQ=PR

QM and RN are the medians

To prove:

(i) ΔNQR = ΔMRQ

(ii) QM = RN

(iii) ΔPMQ = ΔPNR

Solution:

We can prove these by following the given steps-

We know that PQR is an isosceles triangle.

PQ=PR

QM and RN divide PQ and PR into two equal parts

So, PN=NQ=PM=MR (1)

i. In ∆NQR and ∆MRQ,

QR is the common side in both triangles.

Angle Q= Angle R (Angle corresponding to equal sides are equal)

NQ=MR (as per (1))

The triangles are congruent because of the SAS rule.

So, ΔNQR = ΔMRQ

ii. QM and RN are the sides of ∆NQR and ∆MRQ which are congruent.

So, QM=RN (corresponding sides of congruent triangles are equal)

iii. In ΔPMQ and ΔPNR,

Angle P is common in both triangles.

PQ=PR (Isosceles triangle)

PN=PM (as per (1))

The triangles are congruent because of the SAS rule.

So, ΔPMQ = ΔPNR

Therefore, ΔNQR = ΔMRQ, QM=RN, and ΔPMQ = ΔPNR.