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 (1)
Assume points A and B have coordinates a and b on a number line.
Calculate the coordinate of point P as p = 4 3 a + b .
Calculate the coordinate of point Q as q = 4 a + 3 b .
Since p = q unless a = b , points P and Q are different. N o , P and Q are not the same point.
Explanation
Define coordinates of A and B. Let A = a and B = b be the coordinates of points A and B on the number line.
Calculate the coordinate of point P. The coordinate of point P is given by the formula: p = a + 1 + 3 1 ( b − a ) = a + 4 1 ( b − a ) = 4 4 a + b − a = 4 3 a + b So, P is located at 4 3 a + b .
Calculate the coordinate of point Q. The coordinate of point Q is given by the formula: q = b + 1 + 3 1 ( a − b ) = b + 4 1 ( a − b ) = 4 4 b + a − b = 4 3 b + a So, Q is located at 4 a + 3 b .
Compare the coordinates of P and Q. Now, let's compare the coordinates of P and Q :
P = 4 3 a + b and Q = 4 a + 3 b .
If P and Q are the same point, then their coordinates must be equal: 4 3 a + b = 4 a + 3 b Multiplying both sides by 4, we get: 3 a + b = a + 3 b Subtracting a and b from both sides, we get: 2 a = 2 b Dividing both sides by 2, we get: a = b This means that P and Q are the same point only if A and B are the same point. However, the problem states that P partitions the segment from A to B , which implies that A and B are distinct points. Therefore, P and Q are different points.
Conclusion Since P is 4 1 the distance from A to B , and Q is 4 1 the distance from B to A , P and Q are not the same point unless A and B coincide.
Examples
In architecture, when dividing a space or segment according to specific ratios, understanding how different starting points affect the final partitioned locations is crucial. For instance, when designing a building facade, dividing a segment into a 1:3 ratio from left to right versus right to left will yield different aesthetic results. This ensures precise and symmetrical designs are achieved, enhancing the building's visual appeal and structural integrity.
