Bot GPT 4-o mini To find point \( X \) on segment \( \overline{AB} \) that divides \( AB \) in the ratio \( 3:5 \), we can use the section formula. The coordinates of points \( A \) and \( B \) are given as:
- \( A(-1, 2) \)
- \( B(3, 6) \)
If the coordinates of point \( X \) that divides \( \overline{AB} \) in the ratio \( m:n \) are given as:
\[ X \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
where \( m = 3 \) and \( n = 5 \), and \( (x_1, y_1) = (-1, 2) \) and \( (x_2, y_2) = (3, 6) \).
Substituting these values into the formula:
- For the x-coordinate of \( X \):
\[ x = \frac{3 \cdot 3 + 5 \cdot (-1)}{3 + 5} = \frac{9 - 5}{8} = \frac{4}{8} = \frac{1}{2} \]
- For the y-coordinate of \( X \):
\[ y = \frac{3 \cdot 6 + 5 \cdot 2}{3 + 5} = \frac{18 + 10}{8} = \frac{28}{8} = \frac{7}{2} \]
Thus, the coordinates of point \( X \) are:
\[ X\left( \frac{1}{2}, \frac{7}{2} \right) \]
Looking at the responses provided:
- \( (12,312) \)
- \( (125,425) \)
- \( (−14,312) \)
- \( (112,112) \)
None of the options provided match \( X\left( \frac{1}{2}, \frac{7}{2} \right) \).
Therefore, the calculation confirms the coordinates of \( X \) but does not align with any of the response options listed. If you need the result to be in a specific format or any clarification, please let me know!
1 answer