Question

Find the coordinates of the point which divides the line segment joining the points A (-6, 10) and B (3, -8) in the ratio 2 : 7.

Hint:

To avoid mixing up the coordinates, remember the "cross" multiplication: the first part of the ratio (\(m_1\)) multiplies the second point's coordinates (\(x_2, y_2\)).

Answer 1

Step 1: Understanding the Concept:
To find the coordinates of a point dividing a line segment in a given ratio, we use the Section Formula. This formula calculates the weighted average of the x and y coordinates based on the ratio.
Step 2: Key Formula or Approach:
If a point \(P(x, y)\) divides the line segment joining \((x_1, y_1)\) and \((x_2, y_2)\) in the ratio \(m_1 : m_2\), then: \[ P(x, y) = \left( \frac{m_1x_2 + m_2x_1}{m_1 + m_2}, \frac{m_1y_2 + m_2y_1}{m_1 + m_2} \right) \]
Step 3: Detailed Explanation:
1. Given: \(A(-6, 10)\) as \((x_1, y_1)\), \(B(3, -8)\) as \((x_2, y_2)\), and ratio \(m_1 : m_2 = 2 : 7\).
2. Calculate the x-coordinate:
\[ x = \frac{2(3) + 7(-6)}{2 + 7} = \frac{6 - 42}{9} = \frac{-36}{9} = -4 \] 3. Calculate the y-coordinate:
\[ y = \frac{2(-8) + 7(10)}{2 + 7} = \frac{-16 + 70}{9} = \frac{54}{9} = 6 \]
Step 4: Final Answer:
The coordinates of the point are \((-4, 6)\).