Points P, Q, R, S in that order which divide the line segment joining points A(2,5) and B(5,-5) in 5 equal parts. Find the coordinates of P, Q, R and S.

Consider the figure,



Here, use the section formula which states that the coordinates of the point R which divides the line segment joining the two points P(x1,y1) and Q(x2,y2) internally in the ratio m:n is given by mx2+nx1m+n,my2+ny1m+n

Now, the point P divides the line segment AB in the ratio 1:4. So,
m=1n=4x1=2y1=5x2=5y2=-5

Let the coordinates of the point P be (x,y). So, substitute the above values in the section formula.
x=mx2+nx1m+nx=15+421+4x=5+85x=135

Also,
y=my2+ny1m+ny=1-5+451+4y=-5+205y=155y=3

Thus, the coordinates of point P is 135,3
Now, as the line segment is divided into 5 equal parts, so R(x3,y3) is the mid point of PB. So, using the mid point formula we get
x3=x+x22x3=135+52x3=13+2552x3=3810x3=195

Also,
y3=y+y22y3=3-52y3=-22y3=-1

Thus, the coordinates of point R is 195,-1
Similarly, S(x4,y4) is the mid point of RB. So,
x4=x3+x22x4=195+52x4=13+2552x4=4410x4=225

Also,
y4=y3+y22y4=-1-52y4=-62y4=-3

Thus, the coordinates of point S is 225,-3
Similarly, Q(x5,y5) is the mid point of PR. So,
x5=x+x32x5=135+1952x5=13+1952x5=3210x5=165

Also,
y5=y+y32y5=3-12y5=22y5=1

Thus, the coordinates of point Q is 165,1

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