Solving for an endpoint on a directed segment without midpoint given

[agp]

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!

 

[agp]

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'm trying to help a friend's child understand this. So far, we've been able to solve other similar problems with endpoints given. I haven't seen anything like this since 1993. I've tried the section formula with the known values plugged in, treating B like an external & internal point on line AP, nothing has proved correct when plotting on a graph to check answer coordinates. The students have no textbook. We've been to Khan Academy and all over the net searching, but have been unable to find an example like this. Thank you for your time and help.

 

[stapel]

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.

The desired point B is not going to make P a midpoint, but the Midpoint Formula (here) is suggestive of the way you want to go.

Instead of P splitting the segment into halves, the split is off to one side of the center:

A           P     B
*-----------*-----*

But do you see that you can have P related to a midpoint of a section of the line segment?

A     C     P     B
*-----*-----*-----*

Since P splits the line into a two-to-one ratio, you've got AB split by P into a short segment, BP, and a twice-as-long segment, AP. Since this segment is indeed twice as long, then there is a point (which I've called "C" above) which is the midpoint of the segment AP. Can you find the coordinates of C?

Then P will be the midpoint of the segment CB. Can you plug C and P into the Midpoint Formula, and then solve for B?

 

[pka]

I'm trying to help a friend's child understand this. So far, we've been able to solve other similar problems with endpoints given. I haven't seen anything like this since 1993.

You should have told that to begin with. Had you been a student, we would want to just hand out a solution.

The line segment $\displaystyle \overline{AB}=(t[x_2-4]+4,t[y_2-6]+6),~0\le t\le 1.$

We can see that if $\displaystyle t=0$ we get point $\displaystyle A$ and if $\displaystyle t=1$ we get point $\displaystyle B$

The point you want is when $\displaystyle \frac{2}{3}$.

 

[Ishuda]

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)

 

[Deleted member 4993]

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!

Another way: (assuming P lies on the line AB)

Can you calculate the length of PB in terms of (x₂, y₂)? (= d₂)

Can you calculate the length of PA? (= d₁)

Can you calculate the length of AB? (= d₃)

Now you have:

2*d₂ = d₁ ............................ (1)

And

d₃ = d₂ + d₁............................ (2)

Two equations and two unknowns - solve it......