Point A is located at (0, 4) and point B is located at (−2, −3). Find the x value for the point that is 1/4 the distance from point A to point B. −1 −0.75 −0.5 −0.25,

- First thing, find the distance between A and B:
 
AB =[tex] \sqrt{( x_{2} - x_{1})^{2} + ( y_{2} - y_{1})^{2} } [/tex] =
      =[tex] \sqrt{(0-(-2))^{2} + (4-(-3))^{2} } [/tex] =
      = [tex] \sqrt{4+49} [/tex] =
      = √53

- Hence, we know that the distance between point A and the point C to be found is: 
AC = √53 / 4

- The only other thing we know about point C is that it lays on the line that connects A and B. Let's use the point-slope formula to find the equation of this line:

y - y₂ = [tex] \frac{y_{2} - y_{1} }{x_{2} - x_{1} }[/tex] (x - x₂)
y - (-3) = (7/2)(x - (-2))
y + 3 = (7/2)(x + 2)
y = (7/2)x + 7 - 3
y = (7/2)x + 4

This is the equation that ties the coordinates of every point laying on it, therefore it is vaid for A, B and, above all, C which will be:
C (x , (7/2)x + 4)

- Now let's find the distnce between A and C:
AC =[tex] \sqrt{( x_{2} - x_{1})^{2} + ( y_{2} - y_{1})^{2} } [/tex]=
      =[tex] \sqrt{(x-0)^{2} + ((7/2)x+4-4)^{2} } [/tex] =
      =[tex]\sqrt{x^{2} + (49/4)x^{2} }[/tex] =
      =[tex] \sqrt{(53/4) x^{2} } [/tex] =
      =+/-(√53/2)x 
- Lastly, we know this distance must be equal to √53 / 4, therefore you need to set and solve the equation:
+/-(√53/2)x = √53 / 4
(1/2)x = +/-(1/4)
x = +/-(1/2)

Since point C is towards point B, we have to take the negative answer: 
x = -(1/2)

- Therefore the correct answer is C: -0.5