Question

Find the coordinates of the point which divides the line segment joining \((2, -3)\) and \((7, 9)\) in the ratio \(3:2\).

• A. (5, 4.2)
• B. (5, 8.2)
• C. (4, 10.2)
• D. (3, 12.2)

Hint:

To find the point dividing a segment in ratio \(m:n\), always clarify the order of division and use the formula accordingly.

Answer 1

The Correct Option is A

To find the point that divides the line segment between $(2, -3)$ and $(7, 9)$ in the ratio $3 : 2$, we apply the section formula.

1. The Section Formula:
For a point $P$ dividing the line segment from $A(x_1, y_1)$ to $B(x_2, y_2)$ in the ratio $m : n$, the coordinates of $P$ are:

$ P = \left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right) $

2. Identifying Values:
Given:
$ x_1 = 2, \quad y_1 = -3 $
$ x_2 = 7, \quad y_2 = 9 $
$ m = 3, \quad n = 2 $

3. Calculating the $x$-coordinate:
$ x = \frac{(3 \times 7) + (2 \times 2)}{3 + 2} = \frac{21 + 4}{5} = \frac{25}{5} = 5 $

4. Calculating the $y$-coordinate:
$ y = \frac{(3 \times 9) + (2 \times -3)}{3 + 2} = \frac{27 - 6}{5} = \frac{21}{5} = 4.2 $

5. The Resulting Point:
The coordinates of the dividing point are $ \boxed{(5, 4.2)} $.