First ask yourself what information is this problem giving me, and what is it asking for?
It gives you the two points at the end of the line segment. Whenever you have two points, you also know the distance between them. This is because you have the distance formula:
d = √((x1 - x2)2 + (y1 - y2)2)
If you ever have trouble remembering the distance formula, just remember that it is just another way of writing the pythagorean theorem. You can take any line segment, and unless it is vertical or horizontal, you can draw a right triangle using it as the hypoteneus. If it is vertical, the x values go away, so the distance is just the change in y. Likewise if it is horizontal, the distance is just the change in x.
Also, don't worry too much about which point is (x1,y1) and which point is (x2,y2) - the formula works either way. Just make sure that whichever point you chose as the first point, you use that point's x and y as the first values for both parts.
So in this case, we can say the distance is
√((-3 - -1)2 + (5 - -7)2)
√((-2)2 + (12)2)
√(4 + 144)
√148
Now you could try to put this in a calculator, but it is usually easier to work with a simplified radical. I see this is a multiple of 4, which is a perfect square, so
√(4 * 37)
2√37 ← so that is our distance.
Now we want a point that cuts this distance into two pieces, with one piece being 3 times longer than the other.
x + 3x = 2√37
4x = 2√37
x = 1/2 √37
We also know that this point has to be on the line segment, so the slope from one end point to the other will be the same as the slope from that same endpoint to our new point.
Our slope is rise over run, or (-7 - 5) / (-1 - -3) = -12/2 = -6
Another way of saying this is that for every change in x by 1, y changes by -6. So now we can start to define where this point is. Here, I'm going to use a to refer to some unknown distance.
x? = -3 + a
y? = 5 - 6a
Now we can plug that back into a distance formula with the distance we found before:
1/2 √37 = √((-3 + a - -3)2 + (5 - 6a - 5)2)
The -3 and 5 cancel out, and you get
1/2 √37 = √(a + 36a)
Simplify and sqare both sides
1/4 * 37 = 37a
a = 1/4
Now you can plug a into
x? = -3 + a
y? = 5 - 6a
and get
(-3 + 1/4, 5 - 6/4)
(-11/4 , 7/2) ← And there is one answer. There are actually two answers. If I had used the other endpoint, I would get the other answer.
I know that may have seemed like a lot to follow. But all I did was keep writing down what I knew and/or could extrapolate and keep working towards an answer having faith that all the messy algebra would work out in the end. That's actually the secret to solving most math problems: just keep writing down what you know and keep working towards what you are trying to find.
