Question
Calculate point P
on AB¯¯¯¯¯¯¯¯
such that the ratio of AP
to PB
is 1:3.
(1 point)
Responses
P(−245,−235)
upper P left parenthesis negative Start Fraction 14 over 5 End Fraction comma negative Start Fraction 13 over 5 End Fraction right parenthesis
P(12,112)
upper P left parenthesis Start Fraction 1 over 2 End Fraction comma Start Fraction 3 over 2 End Fraction right parenthesis
P(−212,−214)
upper P left parenthesis negative Start Fraction 5 over 2 End Fraction comma negative Start Fraction 9 over 2 End Fraction right parenthesis
P(−1,−1)
on AB¯¯¯¯¯¯¯¯
such that the ratio of AP
to PB
is 1:3.
(1 point)
Responses
P(−245,−235)
upper P left parenthesis negative Start Fraction 14 over 5 End Fraction comma negative Start Fraction 13 over 5 End Fraction right parenthesis
P(12,112)
upper P left parenthesis Start Fraction 1 over 2 End Fraction comma Start Fraction 3 over 2 End Fraction right parenthesis
P(−212,−214)
upper P left parenthesis negative Start Fraction 5 over 2 End Fraction comma negative Start Fraction 9 over 2 End Fraction right parenthesis
P(−1,−1)
Answers
GPT-4o mini
To find point \( P \) on the line segment \( \overline{AB} \) such that the ratio of \( AP \) to \( PB \) is \( 1:3 \), we can use the section formula.
Let's denote the coordinates of points \( A \) and \( B \):
- Let \( A(x_1, y_1) \)
- Let \( B(x_2, y_2) \)
Given the ratio of \( AP \) to \( PB \) is \( 1:3 \), it means that point \( P \) divides the segment \( \overline{AB} \) in the ratio \( m:n = 1:3 \).
The section formula tells us that the coordinates of point \( P \) dividing the segment \( \overline{AB} \) in the ratio \( m:n \) are given by:
\[
P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]
Using \( m = 1 \) and \( n = 3 \), we have:
\[
P\left(\frac{1 \cdot x_2 + 3 \cdot x_1}{1 + 3}, \frac{1 \cdot y_2 + 3 \cdot y_1}{1 + 3}\right) = P\left(\frac{x_2 + 3x_1}{4}, \frac{y_2 + 3y_1}{4}\right)
\]
To specifically calculate the coordinates of \( P \), the coordinates of \( A \) and \( B \) must be known. Since no specific coordinates have been provided in your question, we cannot calculate the exact location of \( P \).
If you have coordinates for points \( A \) and \( B \), please provide them so that we can proceed with the calculation. Otherwise, based on the options given, we could also check if any of the provided points \( P \) satisfies the condition depending on some possible hypothetical locations for \( A \) and \( B \).
Could you provide coordinates for points \( A \) and \( B \) or clarify the problem statement?
Let's denote the coordinates of points \( A \) and \( B \):
- Let \( A(x_1, y_1) \)
- Let \( B(x_2, y_2) \)
Given the ratio of \( AP \) to \( PB \) is \( 1:3 \), it means that point \( P \) divides the segment \( \overline{AB} \) in the ratio \( m:n = 1:3 \).
The section formula tells us that the coordinates of point \( P \) dividing the segment \( \overline{AB} \) in the ratio \( m:n \) are given by:
\[
P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]
Using \( m = 1 \) and \( n = 3 \), we have:
\[
P\left(\frac{1 \cdot x_2 + 3 \cdot x_1}{1 + 3}, \frac{1 \cdot y_2 + 3 \cdot y_1}{1 + 3}\right) = P\left(\frac{x_2 + 3x_1}{4}, \frac{y_2 + 3y_1}{4}\right)
\]
To specifically calculate the coordinates of \( P \), the coordinates of \( A \) and \( B \) must be known. Since no specific coordinates have been provided in your question, we cannot calculate the exact location of \( P \).
If you have coordinates for points \( A \) and \( B \), please provide them so that we can proceed with the calculation. Otherwise, based on the options given, we could also check if any of the provided points \( P \) satisfies the condition depending on some possible hypothetical locations for \( A \) and \( B \).
Could you provide coordinates for points \( A \) and \( B \) or clarify the problem statement?