We have to find the lengths of the medians of a triangle whose co-ordinates of the vertices are A (0,-1); B (2, 1) and C (0, 3)

So we should find the mid-points of the sides of the triangle.

In general to find the mid-point P(x,y) of two points

A

(

x

1

,

y

1

)

A(x1,y1) and

B

(

x

2

,

y

2

)

B(x2,y2) we use section FORMULA as,

P

(

x

,

y

)

=

(

x

1

+

x

2

2

,

y

1

+

y

2

2

)

P(x,y)=(x1+x22,y1+y22)

Therefore mid-point P of side AB can be written as,

P

(

x

,

y

)

=

(

2

+

0

2

,

1

−

1

2

)

P(x,y)=(2+02,1-12)

Now EQUATE the individual terms to get,

x = 1

y = 0

So co-ordinates of P is (1, 0)

Similarly mid-point Q of side BC can be written as

Q

(

x

,

y

)

=

(

2

+

0

2

,

3

+

1

2

)

Q(x,y)=(2+02,3+12)

Now equate the individual terms to get,

x = 1

y = 2

So co-ordinates of Q is (1, 2)

Similarly mid-point R of side AC can be written as,

R

(

x

,

y

)

=

(

0

+

0

2

,

3

−

1

2

)

R(x,y)=(0+02,3-12)

Now equate the individual terms to get,

x =1

y= 2

So co-ordinates of Q is (1, 2)

Similarly mid-point R of side AC can be written as,

R

(

x

,

y

)

=

(

0

+

0

2

,

3

−

1

2

)

R(x,y)=(0+02,3-12)

Now equate the individual terms to get,

x = 1

y = 1

So co-ordinates of R is (0, 1)

Therefore LENGTH of median from A to the side BC is,

A

Q

=

√

(

0

−

1

)

2

+

(

−

1

−

2

)

2

AQ=(0-1)2+(-1-2)2

=

√

1

+

9

=1+9

=

√

10

=10