Question

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 $\frac{1}{4}$ the distance from one endpoint to the other. no, p is $\frac{1}{4}$ the distance from a to b, and q is $\frac{1}{4}$ the distance from b to a. no, q is closer to a and p is closer to b.

Explanation:

Step1: Analyze point P

Point P partitions the segment from A to B in a 1:3 ratio. So P is $\frac{1}{1 + 3}=\frac{1}{4}$ of the distance from A to B.

Step2: Analyze point Q

Point Q partitions the segment from B to A in a 1:3 ratio. So Q is $\frac{1}{1+3}=\frac{1}{4}$ of the distance from B to A.

Answer:

C. No, P is $\frac{1}{4}$ the distance from A to B, and Q is $\frac{1}{4}$ the distance from B to A.