Question

A point P divides the line segment joining A(2, 3) and B(8, 9) in the ratio 1:2. What are the coordinates of P?

• A. (4, 5)
• B. (6, 7)
• C. (3, 4)
• D. (5, 6)

Hint:

Apply the section formula systematically and verify the result lies on the line segment for accuracy.

Answer 1

The Correct Option is A

To find the coordinates of point P dividing the line segment AB in the ratio 1:2, we use the section formula:

1. The section formula for a point dividing a segment joining \( (x_1, y_1) \) and \( (x_2, y_2) \) in ratio \( m:n \) is: \[ \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right). \]
2. Given: A(2, 3), B(8, 9), ratio 1:2 (\( m = 1 \), \( n = 2 \)).
3. Calculate the x-coordinate: \[ x = \frac{1 \times 8 + 2 \times 2}{1 + 2} = \frac{8 + 4}{3} = \frac{12}{3} = 4. \]
4. Calculate the y-coordinate: \[ y = \frac{1 \times 9 + 2 \times 3}{1 + 2} = \frac{9 + 6}{3} = \frac{15}{3} = 5. \]
5. The coordinates of P are (4, 5).
6. Match with options: (4, 5) corresponds to option (A).

Thus, the correct answer is: \[ \boxed{(4, 5)} \]