Question
Calculate point P on AB¯¯¯¯¯¯¯¯ such that the ratio of AP to PB is 1:3.(1 point)ResponsesP(−212,−214)upper P left parenthesis negative Start Fraction 5 over 2 End Fraction comma negative Start Fraction 9 over 2 End Fraction right parenthesisP(−245,−235)upper P left parenthesis negative Start Fraction 14 over 5 End Fraction comma negative Start Fraction 13 over 5 End Fraction right parenthesisP(12,112)upper P left parenthesis Start Fraction 1 over 2 End Fraction comma Start Fraction 3 over 2 End Fraction right parenthesisP(−1,−1)upper P left parenthesis negative 1 comma negative 1 right parenthesis
Answers
GPT-4o mini
To find point \( P \) on segment \( \overline{AB} \) such that the ratio of \( AP \) to \( PB \) is \( 1:3 \), we can apply the section formula.
Let’s denote:
- Point \( A \) as \( A(x_1, y_1) \)
- Point \( B \) as \( B(x_2, y_2) \)
- Point \( P \) will be at position \( \left( x, y \right) \)
According to the section formula, if a point \( P \) divides the line segment \( AB \) in the ratio \( m:n \), then the coordinates of \( P \) are given by:
\[
P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]
In our case, since the ratio \( AP:PB = 1:3 \), we have \( m = 1 \) and \( n = 3 \).
Now, we need to confirm the coordinates of points \( A \) and \( B \). Unfortunately, these coordinates have not been provided in your question.
Let’s assume some example coordinates for point \( A \) and point \( B \):
- \( A = (x_1, y_1) = (-2, -2) \)
- \( B = (x_2, y_2) = (2, 2) \)
Using the section formula:
\[
P\left(\frac{1 \cdot 2 + 3 \cdot (-2)}{1+3}, \frac{1 \cdot 2 + 3 \cdot (-2)}{1+3}\right)
\]
Calculating for x-coordinate:
\[
P_x = \frac{2 - 6}{4} = \frac{-4}{4} = -1
\]
Calculating for y-coordinate:
\[
P_y = \frac{2 - 6}{4} = \frac{-4}{4} = -1
\]
Thus, \( P = (-1, -1) \).
Now looking at your response options, we see:
**P(-1, -1)** is one of the options given. Hence this choice satisfies the requirement provided the assumed coordinates of points \( A \) and \( B \) are correct.
If you provide actual coordinates for \( A \) and \( B \), I can recalculate accordingly. Otherwise, based on this calculation, point \( P \) would be:
**P(-1, -1)**.
Let’s denote:
- Point \( A \) as \( A(x_1, y_1) \)
- Point \( B \) as \( B(x_2, y_2) \)
- Point \( P \) will be at position \( \left( x, y \right) \)
According to the section formula, if a point \( P \) divides the line segment \( AB \) in the ratio \( m:n \), then the coordinates of \( P \) are given by:
\[
P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]
In our case, since the ratio \( AP:PB = 1:3 \), we have \( m = 1 \) and \( n = 3 \).
Now, we need to confirm the coordinates of points \( A \) and \( B \). Unfortunately, these coordinates have not been provided in your question.
Let’s assume some example coordinates for point \( A \) and point \( B \):
- \( A = (x_1, y_1) = (-2, -2) \)
- \( B = (x_2, y_2) = (2, 2) \)
Using the section formula:
\[
P\left(\frac{1 \cdot 2 + 3 \cdot (-2)}{1+3}, \frac{1 \cdot 2 + 3 \cdot (-2)}{1+3}\right)
\]
Calculating for x-coordinate:
\[
P_x = \frac{2 - 6}{4} = \frac{-4}{4} = -1
\]
Calculating for y-coordinate:
\[
P_y = \frac{2 - 6}{4} = \frac{-4}{4} = -1
\]
Thus, \( P = (-1, -1) \).
Now looking at your response options, we see:
**P(-1, -1)** is one of the options given. Hence this choice satisfies the requirement provided the assumed coordinates of points \( A \) and \( B \) are correct.
If you provide actual coordinates for \( A \) and \( B \), I can recalculate accordingly. Otherwise, based on this calculation, point \( P \) would be:
**P(-1, -1)**.