SOLUTION: Three corners of a parallelogram are P(1,-1), Q(-3,4), R(6,10). The fourth corner, S, is diagonally opposite P. Find the coordinates of S.

Algebra ->  Parallelograms -> SOLUTION: Three corners of a parallelogram are P(1,-1), Q(-3,4), R(6,10). The fourth corner, S, is diagonally opposite P. Find the coordinates of S.      Log On


   

Question 212834: Three corners of a parallelogram are P(1,-1), Q(-3,4), R(6,10). The fourth corner, S, is diagonally opposite P. Find the coordinates of S.
Answer by algebrapro18(249) About Me  (Show Source):
You can put this solution on YOUR website!
If you graph those points you see that PQ is parallel to RS and QR is parallel to PS. And that PS and RS intersect at the point we want, S. And we know that parallel lines have the same slope, so we can find the equations of lines RS and PS and then solve the system to find point S.

So we know that the slope of QR is the same as PS. We will call that slope M1. Now using the formula for slope we find M1.

M1+=+%2810-4%29%2F%286%2B3%29+=+6%2F9+=+2%2F3

So we have a point on the line PS, namely (1,-1), and its slope so we can use the point slope formula to find the equation of the line.

y-y1=m%28x-x1%29
y%2B1=%282%2F3%29%28x-1%29
y=%282%2F3%29x-2%2F3-1
y=%282%2F3%29x-5%2F3

Now we find the slope of QP, which we will call M2.

M2+=+%284%2B1%29%2F%28-3-1%29+=+5%2F-4+=+-5%2F4

So we have a point on the line RS, namely (6,10), and its slope so we can use the point slope formula to find the equation of the line.
y-y1=m%28x-x1%29
y-10=%28-5%2F4%29%28x-6%29
y=%28-5%2F4%29x%2B15%2F2%2B10
y=%28-5%2F4%29x%2B35%2F2
So we have 2 equations in 2 unknowns so we can solve the system of equations. Since both equations have been solved for y we can set them equal to eachother and solve for x.
%28-5%2F4%29x%2B35%2F2+=+%282%2F3%29x-5%2F3
Since I don't like fractions I multiplied the entire equation by 12
12(%28-5%2F4%29x%2B35%2F2+=+%282%2F3%29x-5%2F3)
-15x%2B210+=+8x-20
210+=+23x-20
230+=+23x
x = 10
now we plug into one of the equations and solve for y. Lets pick the equation for RS.
y=%28-5%2F4%29x%2B35%2F2
y=%28-5%2F4%29%2810%29%2B35%2F2
y=-25%2F2%2B35%2F2
y = 5
so S(10,5).