Let P be the point (0,-1) and Q be the point (3,2). In this Cartesian coordinate system, find the point on the line segment PQ that is twice as far from P as from Q.

Question

Let P be the point (0,-1) and Q be the point (3,2). In this Cartesian coordinate system, find the point on the line segment PQ that is twice as far from P as from Q.
Please help, I need this for a test and i have totally forgotten the formula and notes
 and how to do anything regarding this.

Reiny · Answer

I am sure you made a sketch
Let M(x,y) be that point, so that PM:MQ = 2:1

complete the two right-angled triangles with PM and PQ as the hypotenuse.
by ratios:
(x-0)/(3-0) = 2/3
3x=6
x=2

(y+1)/(2+1)=2/3
3y+3=6
3y=3
y=1

your point is (2,1)

check:
PM = sqrt(2^2+2^2)= sqrt(8) = 2sqrt(2)
MQ = sqrt(1^2 + 1^2) = sqrt(2)
thus PM = 2MQ , as required