Question

In the adjoining figure, ∆ is an isosceles triangle in which PQ=PR and L, M, N are the (3) midpoints of QR, RP and PQ respectively. Show that PL ⊥ nd PL is bisected by MN

Answer 1

Answer:

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Using mid-point theorem, 

we have

MN∥PQ and MN=21PQ⇒MN=PL

Similarly, we have

LM=PN

In triangles NML and LPN, we have

MN=PL

LM=PN

and, LN=NL

So, by SSS congruence criterion, we obtain

ΔNML≅ΔLPN

⇒∠MNL=∠PLN and ∠MLN=∠LNP

⇒∠MNL=∠LNP=∠PLM=∠MLN

⇒∠PNM=∠PLM

∴LN=MN

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Answer 2

Answer:

[tex]

Using mid-point theorem, 

we have

MN∥PQ and MN=21PQ⇒MN=PL

Similarly, we have

LM=PN

In triangles NML and LPN, we have

MN=PL

LM=PN

and, LN=NL

So, by SSS congruence criterion, we obtain

ΔNML≅ΔLPN

⇒∠MNL=∠PLN and ∠MLN=∠LNP

⇒∠MNL=∠LNP=∠PLM=∠MLN

⇒∠PNM=∠PLM

∴LN=MN

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