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(−1,−1) upper P left parenthesis negative 1 comma negative 1 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 Skip to navigation
Answers
GPT-4o mini
To find the point \( P \) on line segment \( \overline{AB} \) such that the ratio \( \frac{AP}{PB} = \frac{1}{3} \), we can use the formula for finding a point that divides a segment in a given ratio.
Let points \( A \) and \( B \) be defined as:
- \( A(x_1, y_1) \)
- \( B(x_2, y_2) \)
We want to find point \( P \) that divides \( \overline{AB} \) in the ratio \( m:n = 1:3 \).
Using the section formula, the coordinates of point \( P \) can be calculated as follows:
\[
P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]
Substituting \( m = 1 \) and \( n = 3 \):
\[
P\left(\frac{1 \cdot x_2 + 3 \cdot x_1}{1+3}, \frac{1 \cdot y_2 + 3 \cdot y_1}{1+3}\right)
\]
This simplifies to:
\[
P\left(\frac{x_2 + 3x_1}{4}, \frac{y_2 + 3y_1}{4}\right)
\]
To proceed with the calculation, we need the coordinates of points \( A \) and \( B \). However, since their coordinates are not provided in your initial question, please provide the coordinates of points \( A \) and \( B \) so we can determine the coordinates of point \( P \).
Let points \( A \) and \( B \) be defined as:
- \( A(x_1, y_1) \)
- \( B(x_2, y_2) \)
We want to find point \( P \) that divides \( \overline{AB} \) in the ratio \( m:n = 1:3 \).
Using the section formula, the coordinates of point \( P \) can be calculated as follows:
\[
P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]
Substituting \( m = 1 \) and \( n = 3 \):
\[
P\left(\frac{1 \cdot x_2 + 3 \cdot x_1}{1+3}, \frac{1 \cdot y_2 + 3 \cdot y_1}{1+3}\right)
\]
This simplifies to:
\[
P\left(\frac{x_2 + 3x_1}{4}, \frac{y_2 + 3y_1}{4}\right)
\]
To proceed with the calculation, we need the coordinates of points \( A \) and \( B \). However, since their coordinates are not provided in your initial question, please provide the coordinates of points \( A \) and \( B \) so we can determine the coordinates of point \( P \).
Use the image to answer the question. Determine the coordinates of point J on AB¯¯¯¯¯¯¯¯ so that the ratio of AJ to JB is 1:1. (1 point) Responses J(0,2) upper J left parenthesis 0 comma 2 right parenthesis J(0,3) upper J left parenthesis 0 comma 3 right parenthesis J(0,−3) upper J left parenthesis 0 comma negative 3 right parenthesis J(3,0)
GPT-4o mini
To find the coordinates of point \( J \) on line segment \( \overline{AB} \) such that the ratio \( \frac{AJ}{JB} = 1:1 \), we need to find the midpoint of segment \( \overline{AB} \).
The midpoint \( J \) can be calculated using the following formula:
\[
J\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\]
where \( (x_1, y_1) \) are the coordinates of point \( A \) and \( (x_2, y_2) \) are the coordinates of point \( B \).
Since the coordinates of points \( A \) and \( B \) were not provided in your question, I cannot provide the exact coordinates for point \( J \). However, when the ratio is \( 1:1 \), the point \( J \) will always be the midpoint of \( \overline{AB} \).
Based on the options you've presented:
- \( J(0, 2) \)
- \( J(0, 3) \)
- \( J(0, -3) \)
- \( J(3, 0) \)
If you have access to the image you mentioned, look for the midpoint of \( A \) and \( B \), which will be \( J \).
If you can provide the coordinates of points \( A \) and \( B \), I can compute the precise coordinates for point \( J \). Otherwise, you can simply select the coordinate that you believe corresponds the midpoint of whatever points \( A \) and \( B \) are displayed in your provided image.
The midpoint \( J \) can be calculated using the following formula:
\[
J\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\]
where \( (x_1, y_1) \) are the coordinates of point \( A \) and \( (x_2, y_2) \) are the coordinates of point \( B \).
Since the coordinates of points \( A \) and \( B \) were not provided in your question, I cannot provide the exact coordinates for point \( J \). However, when the ratio is \( 1:1 \), the point \( J \) will always be the midpoint of \( \overline{AB} \).
Based on the options you've presented:
- \( J(0, 2) \)
- \( J(0, 3) \)
- \( J(0, -3) \)
- \( J(3, 0) \)
If you have access to the image you mentioned, look for the midpoint of \( A \) and \( B \), which will be \( J \).
If you can provide the coordinates of points \( A \) and \( B \), I can compute the precise coordinates for point \( J \). Otherwise, you can simply select the coordinate that you believe corresponds the midpoint of whatever points \( A \) and \( B \) are displayed in your provided image.