Question
Calculate point P on AB such that the ratio of AP to PB is 1:3.
(1 point)
0 (,12)
◎ P（-2片，-2号）
O P(-2층,-2룸)
• P(-1, -1)
(1 point)
0 (,12)
◎ P（-2片，-2号）
O P(-2층,-2룸)
• P(-1, -1)
Answers
GPT-4o mini
To find point \( P \) on line segment \( AB \) such that the ratio of \( AP \) to \( PB \) is \( 1:3 \), we can use the section formula.
Let the coordinates of points \( A \) and \( B \) be \( A(x_1, y_1) \) and \( B(x_2, y_2) \). The coordinates of point \( P \) that divides the line segment \( AB \) in the ratio \( m:n \) can be calculated using the formula:
\[
P\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\]
In this case, since the ratio \( AP:PB = 1:3 \), we set \( m = 1 \) and \( n = 3 \).
Now, we need the coordinates of points \( A \) and \( B \). However, we will derive point \( P \) based on this ratio itself without specific coordinates.
Given that \( P \) divides \( AB \) in the ratio \( 1:3 \):
This means \( P \) is closer to point \( A \). Specifically, \( P \) can be found using the parameter \( k \) representing the total parts as \( m+n = 1 + 3 = 4 \).
Thus, the coordinates of \( P \) (as a fraction of the distance from \( A \)) can be expressed as:
\[
P = A + \frac{1}{4}(B - A)
\]
This means \( P \) is three-fourths (\( \frac{3}{4} \)) of the way along segment \( AB \) from point \( A \) to point \( B \).
Since we don't have the specific coordinates of \( A \) and \( B \), we can't derive the exact position of \( P \). However, you provide some options for \( P \):
1. \( P(0, 12) \)
2. \( P(-2, -2) \)
3. \( P(-2, -2) \)
4. \( P(-1, -1) \)
Among these options, without loss of generality, we would need to compare how these points ratio their distances against an assumed \( A \) and \( B \). Since we don't have definite coordinates for \( A \) and \( B \) the exact computation isn't possible here.
If coordinates \( A \) and \( B \) are provided, please resend it for precise computation.
In the absence of specific coordinates, side note that the point \( P(-2, -2) \) can occur if it corresponds within selected assumptions. So if you want the average confirmation on values, solid definitions on points \( A \) and \( B \) are essential. Please provide them if you seek a detailed computation.
Let the coordinates of points \( A \) and \( B \) be \( A(x_1, y_1) \) and \( B(x_2, y_2) \). The coordinates of point \( P \) that divides the line segment \( AB \) in the ratio \( m:n \) can be calculated using the formula:
\[
P\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\]
In this case, since the ratio \( AP:PB = 1:3 \), we set \( m = 1 \) and \( n = 3 \).
Now, we need the coordinates of points \( A \) and \( B \). However, we will derive point \( P \) based on this ratio itself without specific coordinates.
Given that \( P \) divides \( AB \) in the ratio \( 1:3 \):
This means \( P \) is closer to point \( A \). Specifically, \( P \) can be found using the parameter \( k \) representing the total parts as \( m+n = 1 + 3 = 4 \).
Thus, the coordinates of \( P \) (as a fraction of the distance from \( A \)) can be expressed as:
\[
P = A + \frac{1}{4}(B - A)
\]
This means \( P \) is three-fourths (\( \frac{3}{4} \)) of the way along segment \( AB \) from point \( A \) to point \( B \).
Since we don't have the specific coordinates of \( A \) and \( B \), we can't derive the exact position of \( P \). However, you provide some options for \( P \):
1. \( P(0, 12) \)
2. \( P(-2, -2) \)
3. \( P(-2, -2) \)
4. \( P(-1, -1) \)
Among these options, without loss of generality, we would need to compare how these points ratio their distances against an assumed \( A \) and \( B \). Since we don't have definite coordinates for \( A \) and \( B \) the exact computation isn't possible here.
If coordinates \( A \) and \( B \) are provided, please resend it for precise computation.
In the absence of specific coordinates, side note that the point \( P(-2, -2) \) can occur if it corresponds within selected assumptions. So if you want the average confirmation on values, solid definitions on points \( A \) and \( B \) are essential. Please provide them if you seek a detailed computation.