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
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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
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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
[/tex]
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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