# The vertices of a triangle are A(-27, 24), B(9, −12) and C(18, −3). Let D be the co-ordinates of the point where the median from C meets the altitude from B. Calculate the length of CD.

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The vertices of a triangle are A(-27, 24), B(9, −12) and C(18, −3).

Let D be the co-ordinates of the point where the median from C meets the altitude from B. Calculate the length of CD.

A(-27,24), B(9,-12) and C(18,-3)

Mid point of AB say `M(x_1,y_1),`

`x_1=(-27+9)/2=9`

`y_1=(24-12)/2=6`

Thus we have M(9,6).

Equation of the line joing CM

`y-6=(-3-6)/(18-9)(x-9)`

`y-6=-1(x-9)`

`x+y=15` (i)

Slope of the line AC say `m_1,` where

`m_1=(-3-24)/(18+27)`

`=(-27)/45`

`=-3/5`

So, slope of the line perpendicular to AC will = `-1/((-3/5))=5/3`

Thus equation of line perpendicular to AC and posses through B(9,-12) is

`y+12=(5/3)(x-9)`

`3y+36=5x-45`

`-5x+3y=-81` (ii)

To obtain coordinates of point D, solve (i) and (ii). Thus

y=-3/4 and x=63/4 .

So the distance between D(63/4,-3/4) and A(-27,24) is

`=sqrt((-27-63/4)^2+(24+3/4)^2)`

`=sqrt((171/4)^2+(99/4)^2)`

`=sqrt(39042/16)`

`=49.398` units