To find the coordinates of point \( X \) on segment \( AB \) such that the ratio \( AX:XB = 3:5 \), we can use the section formula. The section formula states that if a point \( X \) divides the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then the coordinates of point \( X \) are given by:
\[ X\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
For your problem:
- \( A(-1, 2) \) with coordinates \( (x_1, y_1) = (-1, 2) \)
- \( B(3, 6) \) with coordinates \( (x_2, y_2) = (3, 6) \)
- The ratio \( m:n = 3:5 \), so \( m = 3 \) and \( n = 5 \).
Now, apply the section formula:
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Calculate \( X \) coordinates: \[ X_x = \frac{3(3) + 5(-1)}{3 + 5} = \frac{9 - 5}{8} = \frac{4}{8} = \frac{1}{2} \]
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Calculate \( Y \) coordinates: \[ X_y = \frac{3(6) + 5(2)}{3 + 5} = \frac{18 + 10}{8} = \frac{28}{8} = \frac{7}{2} \]
So the coordinates of point \( X \) are \( \left( \frac{1}{2}, \frac{7}{2} \right) \).
None of your responses exactly matches the found coordinates, but it appears there may have been some misunderstanding or misrepresentation of the options. However, \( \left( \frac{1}{2}, \frac{7}{2} \right) \) is the correct point that divides \( AB \) in the specified ratio of \( 3:5 \).