Question
Determine the coordinates of point P on AB¯¯¯¯¯¯¯¯ such that the ratio of AP to PB is 1:3.(1 point)P is located at (,).
Answers
GPT-4o mini
To find the coordinates of point \( P \) on line segment \( \overline{AB} \) such that the ratio of \( AP \) to \( PB \) is \( 1:3 \), you can use the section formula.
1. Let the coordinates of point \( A \) be \( (x_1, y_1) \) and point \( B \) be \( (x_2, y_2) \).
2. The ratio of \( AP \) to \( PB \) is given as \( 1:3 \), which means \( AP = 1k \) and \( PB = 3k \) for some \( k \).
3. The total ratio is \( 1 + 3 = 4 \).
Using the section formula, the coordinates of point \( P \) (which divides \( \overline{AB} \) in the ratio \( m:n \), where \( m = 1 \) and \( n = 3 \)) can be calculated as:
\[
P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right).
\]
Plugging in the values of \( m \) and \( n \):
\[
P\left(\frac{1 \cdot x_2 + 3 \cdot x_1}{1 + 3}, \frac{1 \cdot y_2 + 3 \cdot y_1}{1 + 3}\right) = P\left(\frac{x_2 + 3x_1}{4}, \frac{y_2 + 3y_1}{4}\right).
\]
If you know the specific coordinates of points \( A \) and \( B \), substitute those values into the formula above to find the coordinates of point \( P \). If you provide the coordinates of points \( A \) and \( B \), I can calculate it for you.
1. Let the coordinates of point \( A \) be \( (x_1, y_1) \) and point \( B \) be \( (x_2, y_2) \).
2. The ratio of \( AP \) to \( PB \) is given as \( 1:3 \), which means \( AP = 1k \) and \( PB = 3k \) for some \( k \).
3. The total ratio is \( 1 + 3 = 4 \).
Using the section formula, the coordinates of point \( P \) (which divides \( \overline{AB} \) in the ratio \( m:n \), where \( m = 1 \) and \( n = 3 \)) can be calculated as:
\[
P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right).
\]
Plugging in the values of \( m \) and \( n \):
\[
P\left(\frac{1 \cdot x_2 + 3 \cdot x_1}{1 + 3}, \frac{1 \cdot y_2 + 3 \cdot y_1}{1 + 3}\right) = P\left(\frac{x_2 + 3x_1}{4}, \frac{y_2 + 3y_1}{4}\right).
\]
If you know the specific coordinates of points \( A \) and \( B \), substitute those values into the formula above to find the coordinates of point \( P \). If you provide the coordinates of points \( A \) and \( B \), I can calculate it for you.