Point P partitions the directed segment from A to B into a 1:3 ratio. Q partitions the directed segment from B to A into a 1:3 ratio. Are P and Q the same point? Why or why not?Yes, they both partition the segment into a 1:3 ratio.Yes, they are both the distance from one endpoint to the other.No, P is the distance from A to B, and Q is the distance from B to A.No, Q is closer to A and P is closer to B.

Answer with explanation:It is given that ,Point P partitions the directed segment from A to B into a 1:3 ratio. Q partitions the directed segment from B to A into a 1:3 ratio.Let coordinates of A be (a,0) and B be (0,b).And Coordinate of Point P be (x,y) and coordinate of Q be (p,q).We will use section formula to determine whether P and Q are the same point or not.Formula for external Division[tex]x=\frac{m x_{2}+n x_{1}}{m+n}\\\\y=\frac{my_{2}+n y_{1}}{m+n}\\\\x=\frac{1\times 0+3\times a}{1+3}\\\\x=\frac{3a}{4}\\\\y=\frac{1\times b+3\times 0}{1+3}\\\\y=\frac{b}{4}\\\\p=\frac{1\times a+3\times 0}{1+3}\\\\p=\frac{a}{4}\\\\q=\frac{1\times 0+3\times b}{1+3}\\\\x=\frac{3 b}{4} [/tex]Coordinates of P and Q are       [tex]=P(\frac{a}{4},\frac{3b}{4}),Q(\frac{3b}{4},\frac{a}{4})[/tex]So, P and Q are not the same point.Option D: No, P is closer to A , and Q is closer from B .