Question

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.

Answer 1

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

Explanation:

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)`