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(−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) upper P left parenthesis negative 1 comma negative 1 right parenthesis P(12,112)
Answers
GPT-4o mini
To find point \( P \) on line segment \( \overline{AB} \) such that the ratio \( AP : PB = 1 : 3 \), we can use the section formula. According to the section formula, if a point \( P \) divides the line segment \( \overline{AB} \) in the ratio \( m:n \), then the coordinates of point \( P \) can be calculated using
\[
P\left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right)
\]
Here, we have \( m = 1 \) and \( n = 3 \).
Assuming point \( A \) has coordinates \( A(x_1, y_1) \) and point \( B \) has coordinates \( B(x_2, y_2) \), we can compute the coordinates of point \( P \) using the equation above.
To provide specific coordinates for points \( A \) and \( B \), we’d need their values. However, using the fact that \( AP:PB = 1:3 \), point \( P \) would be located one-fourth of the distance from \( A \) to \( B \).
1. **Determine coordinates \( A \) and \( B \)**: Let's assume examples of \( A \) and \( B \) for calculations.
For example,
- Let \( A = (x_1, y_1) = (−1, −1) \)
- Let \( B = (x_2, y_2) = (1, 1) \)
2. **Calculate \( P \)**: Substituting into the formula:
\[
P\left( \frac{1 \cdot 1 + 3 \cdot (-1)}{1 + 3}, \frac{1 \cdot 1 + 3 \cdot (-1)}{1 + 3} \right) = P\left( \frac{1 - 3}{4}, \frac{1 - 3}{4} \right) = P\left( \frac{-2}{4}, \frac{-2}{4} \right) = P\left( -\frac{1}{2}, -\frac{1}{2} \right)
\]
3. **Validate against given options**:
Now we can compare this calculated value against the given responses.
The choice \( P(-1, -1) \) is not \( P \) since we calculated \( P \) as \( \left(-\frac{1}{2}, -\frac{1}{2}\right) \), which doesn't match any given point directly.
Since there are no further coordinates provided for points \( A \) and \( B \), it is impossible to provide an explicit value for \( P \) without additional information on \( A \) and \( B \).
If you have specific coordinates for points \( A \) and \( B \), please provide them, and I can execute the calculation with those values instead.
\[
P\left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right)
\]
Here, we have \( m = 1 \) and \( n = 3 \).
Assuming point \( A \) has coordinates \( A(x_1, y_1) \) and point \( B \) has coordinates \( B(x_2, y_2) \), we can compute the coordinates of point \( P \) using the equation above.
To provide specific coordinates for points \( A \) and \( B \), we’d need their values. However, using the fact that \( AP:PB = 1:3 \), point \( P \) would be located one-fourth of the distance from \( A \) to \( B \).
1. **Determine coordinates \( A \) and \( B \)**: Let's assume examples of \( A \) and \( B \) for calculations.
For example,
- Let \( A = (x_1, y_1) = (−1, −1) \)
- Let \( B = (x_2, y_2) = (1, 1) \)
2. **Calculate \( P \)**: Substituting into the formula:
\[
P\left( \frac{1 \cdot 1 + 3 \cdot (-1)}{1 + 3}, \frac{1 \cdot 1 + 3 \cdot (-1)}{1 + 3} \right) = P\left( \frac{1 - 3}{4}, \frac{1 - 3}{4} \right) = P\left( \frac{-2}{4}, \frac{-2}{4} \right) = P\left( -\frac{1}{2}, -\frac{1}{2} \right)
\]
3. **Validate against given options**:
Now we can compare this calculated value against the given responses.
The choice \( P(-1, -1) \) is not \( P \) since we calculated \( P \) as \( \left(-\frac{1}{2}, -\frac{1}{2}\right) \), which doesn't match any given point directly.
Since there are no further coordinates provided for points \( A \) and \( B \), it is impossible to provide an explicit value for \( P \) without additional information on \( A \) and \( B \).
If you have specific coordinates for points \( A \) and \( B \), please provide them, and I can execute the calculation with those values instead.