The length of the median passing through vertex A of a triangle ABC in which AB=c, BC= a, CA=b is
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∵D is the mid-point
∴AB
2
+AC
2
=2[AD
2
+BD
2
]
⇒c
2
+b
2
=2[l
2
+(
2
a
)
2
]
⇒2c
2
+2b
2
=4l
2
+a
2
⇒4l
2
=2b
2
+2c
2
−a
2
⇒4l
2
=b
2
+c
2
+(b
2
+c
2
−a
2
)
⇒4l
2
=b
2
+c
2
+2bccosA (using cosine rule)
⇒4l
2
=(b
2
+c
2
−a
2
)+a
2
+2bccosA
⇒4l
2
=2bccosA+a
2
+2bccosA
⇒4l
2
=4bccosA+a
2
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