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?
A. Yes, they both partition the segment into a $1: 3$ ratio.
B. Yes, they are both $\frac{1}{4}$ the distance from one endpoint to the other.
C. No, $P$ is $\frac{1}{4}$ the distance from $A$ to $B$, and $Q$ is $\frac{1}{4}$ the distance from $B$ to $A$.
D. No, Q is closer to A and P is closer to B.
Answer (2)
Points P and Q are not the same because they divide the segments A B and B A into different parts. Point P is 4 1 the distance from A to B , while point Q is 4 1 the distance from B to A . Thus, the correct answer is C.
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Point P divides segment A B in a 1 : 3 ratio, so its position vector is p = 4 3 a + b .
Point Q divides segment B A in a 1 : 3 ratio, so its position vector is q = 4 a + 3 b .
Comparing the position vectors, p = q only if a = b , meaning A and B are the same point.
Since A and B are distinct, P and Q are different points, with P being 4 1 the distance from A to B and Q being 4 1 the distance from B to A . Therefore, the answer is: N o .
Explanation
Define points A and B. Let A and B be two distinct points in space. We can represent these points using position vectors a and b , respectively.
Find the position vector of point P. Point P divides the directed segment from A to B in a 1 : 3 ratio. This means that the ratio of A P to PB is 1 : 3 . Using the section formula, the position vector of point P , denoted as p , can be expressed in terms of a and b as follows: p = 1 + 3 3 a + 1 b = 4 3 a + b So, p = 4 3 a + b .
Find the position vector of point Q. Point Q divides the directed segment from B to A in a 1 : 3 ratio. This means that the ratio of BQ to Q A is 1 : 3 . Using the section formula, the position vector of point Q , denoted as q , can be expressed in terms of a and b as follows: q = 1 + 3 3 b + 1 a = 4 a + 3 b So, q = 4 a + 3 b .
Compare the position vectors of P and Q. Now, let's compare the position vectors p and q to see if they are the same. If p = q , then P and Q are the same point. Otherwise, they are distinct points. We have: p = 4 3 a + b q = 4 a + 3 b For p and q to be equal, we must have: 4 3 a + b = 4 a + 3 b Multiplying both sides by 4, we get: 3 a + b = a + 3 b Rearranging the terms, we have: 2 a = 2 b a = b This implies that points A and B must be the same point for P and Q to be the same point. However, we are given that P partitions the segment from A to B , which implies that A and B are distinct points. Therefore, P and Q are not the same point.
Conclusion. Since P divides A B in the ratio 1 : 3 , it is 4 1 of the distance from A to B .
Since Q divides B A in the ratio 1 : 3 , it is 4 1 of the distance from B to A .
Therefore, P and Q are not the same point.
Examples
In architecture, when designing a structure, you might need to divide a beam or a space into specific ratios to support weight or create aesthetically pleasing proportions. Understanding how points divide segments in given ratios helps ensure structural integrity and visual harmony in the design.
