Question

In a Δ PQR, if PQ=QR and L, M and N are the mid points of the sides PQ, QR and RP respectively, Prove that LN=MN.

Answer 1

Prove that LN = MN

Explanation:

In ΔPQR,

PQ = QR

Triangle PQR is an isosceles triangle.

Angles opposite to equal sides are equal in triangle.

$\angle \mathrm{QPR}=\angle \mathrm{QRP}$ ----------(1)

L and M are mid-points of PQ and QR respectively.

$\Rightarrow P L=L Q=\frac{P Q}{2}$

$\Rightarrow Q M=M R=\frac{Q R}{2}$

Also, PQ = QR

$\Rightarrow P L=L Q=Q M=M R$

$\Rightarrow \frac{P Q}{2}=\frac{Q R}{2}$ ----------(2)

Consider ΔLPN and ΔMRN,

LP = MR (Using (2))

$\angle \mathrm{LPN}=\angle \mathrm{MRN}$ (using (1))

∠QPR = ∠LPN = ∠QRP = MRN

PN = NR (N is a midpoint of PR)

Therefore, ΔLPN ≅ ΔMRN by SAS congruence criterion

By corresponding parts of congruence triangles are equal.

⇒ LN = MN

Hence proved.

To learn more...

1. In ΔPQR, PQ = QR; L,M and N are the midpoints of the sides of PQ, QR and RP respectively. Prove that LN = MN.

https://brainly.in/question/7390829

2. In triangle pqr if pq=qr and mid points of three sides pq, qr aand rp are l, m and n respectively. prove ln=mn

https://brainly.in/question/768545

Answer 2

Given: In ΔPQR,

PQ = QR

L, M and N are the mid-points of the sides PQ, QR and PR respectively

To prove: LM = MN

Proof: In ΔLPN and ΔMRH,

PN = RN (∵ M is the mid-point of PR)

LP = MR (Half of equal sides)

          ∠P = ∠R (Angle opposite to equal sides)

          ∴ALPN ≅ AMRH (SAS Congruence Criteria)

∴LN = MN (CPCT)