The problem is as follows:
Point P(4,6) lies between points A(10,-12) and B(x2,y2). Find the coordinates of point B so that the directed segment AB is divided into a 2:1 ratio.
I have been able to find P in other problems and solve for ratios but am having trouble with this one. I can find the endpoint with midpoint given, but P is not midpoint, it's 2:1.
Thank you for any help you can provide!
I have found that the easiest way to handle situations such as this is to parameterize the line. As an example, suppose we have a line going through the two points (7, 10) and (10, 16). Then the equation for that line is
y = 2 (x-7) + 10.
We can introduce another variable t [the parametrization variable] so that
x = t + 7
y = 2 t + 10.
That is when t=0 we are at the point (7,10) and when t=3 we are at the point (10,15). By doing this, t is the scaled distance along the line starting at (7,10) and going to (x,y). For example, the distance d(x,y) between (7,10) and (x,y) is given by
d(x,y) = \(\displaystyle \sqrt{(x-7)^2\, +\, (y-10)^2}\, =\, \sqrt{(t+7-7)^2\, +\, (2t+10-10)^2}\, =\, \sqrt{5}\, t\)
Since t=3 gives the point (10,16), the distance between (7,10) and (10,16) is 3 \(\displaystyle \sqrt{5}\). Suppose I wanted to go a distance \(\displaystyle \frac{1}{3}\) of the way down the line from (7,10) toward (10,16). That would mean
d(x,y) = \(\displaystyle \sqrt{5}\, t\) = \(\displaystyle \frac{1}{3}\) d(10,16) = \(\displaystyle \frac{1}{3}\) \(\displaystyle [3\, \sqrt{5}]\, =\, \sqrt{5} \)
or t = 1. Thus (8, 12) is \(\displaystyle \frac{1}{3}\) of the way from (7, 10) to (10, 16)
