Question

Find the co-ordinates of the point which divides line segment joining the points (-1, 7) and (4, -3) in the ratio 2:3 internally.

Hint:

To avoid confusion, always label your points as \((x_1, y_1)\) and \((x_2, y_2)\) and the ratio as \(m_1\) and \(m_2\). Be careful to match \(m_1\) with the coordinates of the second point \((x_2, y_2)\) and \(m_2\) with the coordinates of the first point \((x_1, y_1)\) in the formula.

Answer 1

Step 1: Understanding the Concept:
This problem requires the use of the section formula, which gives the coordinates of a point that divides a line segment into a given ratio.

Step 2: Key Formula or Approach:
The section formula for a point \(P(x, y)\) that divides the line segment joining \(A(x_1, y_1)\) and \(B(x_2, y_2)\) in the ratio \(m_1:m_2\) internally is: \[ x = \frac{m_1x_2 + m_2x_1}{m_1 + m_2} \text{and} y = \frac{m_1y_2 + m_2y_1}{m_1 + m_2} \]

Step 3: Detailed Explanation:
Let the given points be \(A(x_1, y_1) = (-1, 7)\) and \(B(x_2, y_2) = (4, -3)\).
The ratio is \(m_1:m_2 = 2:3\).
So, \(m_1 = 2\) and \(m_2 = 3\).
Calculate the x-coordinate: \[ x = \frac{(2)(4) + (3)(-1)}{2 + 3} = \frac{8 - 3}{5} = \frac{5}{5} = 1 \] Calculate the y-coordinate: \[ y = \frac{(2)(-3) + (3)(7)}{2 + 3} = \frac{-6 + 21}{5} = \frac{15}{5} = 3 \] The coordinates of the point are (1, 3).

Step 4: Final Answer:
The coordinates of the point that divides the line segment are (1, 3).