Bot GPT 4-o mini To find the point \( P \) that is \( \frac{1}{6} \) of the way from point \( A(-2,4) \) to point \( B(4,2) \), we can use the section formula.
The formula for a point \( P \) that divides the segment between points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \) is given by:
\[ P\left(x, y\right) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
In our case, we want to find a point \( P \) that is \( \frac{1}{6} \) of the way from \( A \) to \( B \), which means the ratio \( m:n \) is \( 1:5 \) (because if \( P \) is \( \frac{1}{6} \) of the way from \( A \) to \( B \), it is \( 5/6 \) of the way from \( B \) to \( A \)).
Given:
- \( A(-2, 4) \) \( (x_1, y_1) \)
- \( B(4, 2) \) \( (x_2, y_2) \)
- \( m = 1 \)
- \( n = 5 \)
Now, we can plug the values into the formula:
\[ P_x = \frac{1 \cdot 4 + 5 \cdot (-2)}{1 + 5} = \frac{4 - 10}{6} = \frac{-6}{6} = -1 \]
\[ P_y = \frac{1 \cdot 2 + 5 \cdot 4}{1 + 5} = \frac{2 + 20}{6} = \frac{22}{6} = \frac{11}{3} \approx 3.67 \]
Thus, the coordinates of point \( P \) are approximately \( (-1, 3.67) \).
So, the correct answer is:
(−1, 3.67)
1 answer