SOLUTION: Hello thank you for checking and helping me out. I have a problem for a College Level Analytic Geometry It says: Find the point P such that the line of AP/ the line of AB = r

Question 1078045: Hello thank you for checking and helping me out.
I have a problem for a College Level Analytic Geometry
It says:
Find the point P such that the line of AP/ the line of AB = r
So r = line of AP/ line of AB
If A=(3,4) B=(7,0) r=1/4 Find P.
According to my professor r is the distance or the internal point division.
For me I could not understand what does it mean by AP? and AB?
I asked to further elaborate what it meant my professor said P is x, instead of answers I get more questions.

Found 2 solutions by Fombitz, Boreal:
Answer by Fombitz(32388) (Show Source): You can put this solution on YOUR website!
AB is the distance between A and B.
AP is the distance between A and P.
Find P so that AP is 1/4 of AB.
.
.
.
In rough graphical terms,
A--------------------B
A-----P
.
.
.
So find the x distance between A and B,

Find the y distance between A and B,

Now you only want to do 1/4 of the distance from A to B so starting at A add 1/4 of the distance for both x and y,

.
.
.
Point P is then (4,3)

Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!

The line AB is between (3,4) and (7,0). It's length is sqrt ((0-4)^2+(7-3)^2)) or sqrt (32)=4 sqrt (2).
P is the location of the point, and it is 1/4 of the way from A to B. (1/4)*4 sqrt(2)=sqrt(2).
The distance formula is the sqrt of the x^2 and y^2 distances
Therefore, to get sqrt (2), either x or y has to be [sqrt (2)]^2 and the other 0, OR, each has to be sqrt [(1^2) +(1^2)]=sqrt (2).
Since P is on the line, we need to move from A 1 unit towards B in each of the x and y coordinates. That would make P at (3+1, 4-1) or (4,3).
The line has slope -1, and that is another way of saying the x and y components that make up a slope of -1 are where one is positive 1 and the other -1.

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