How to find the coordinates of point S?

P = (-1, 2), Q = (5, 5), R = (2, -1).

"Find the coordinates of S (S is on the segment QR) if the length of the segment QS is double the length of segment SR.".
Finding this to be tricky. Help is appreciated.

1 Answer
CW
Apr 4, 2018

#S(x,y)=S(3,1)#

Explanation:

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given that #QS=2SR, => QS:SR=2:1#
slope of #RQ=m_(RQ)=(5+1)/(5-2)=6/3=2#
#DeltaQSB and DeltaSRA# are similar,
#=> (QS)/(SB)=(SR)/(RA)#
#=> 2/(5-x)=1/(x-2)#
#=> 2x-4=5-x#,#=> 3x=9, => x=3#
slope of #SQ=(5-y)/(5-x)=M_(RQ)=2#
#=> (5-y)/(5-3)=2, => y=1#
#=> S(x,y)=S(3,1)#

Or use section formula to find #S(x,y)#
If a point #S(x,y)# divides a line segment joining #R(x_1,y_1)and Q(x_2,y_2)# in the ratio #m:n#, i.e., #RS:SQ=m:n#,
then #S(x,y)=((mx_2+nx_1)/(m+n), (my_2+ny_1)/(m+n))#
given #RS:SQ=1:2 and R(2,-1), and Q(5,5)#
#=> S(x,y)=((1*5+2*2)/(1+2), " " (1*5+2*(-1))/(1+2))#
#=> S(x,y)=(9/3, ' "3/3)=(3,1)#

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