The point R divides the line segment AB, where A (- 4, 0) and B (0, 6) such that `R = 3/4 AB`. Find the coordinates of R

Answer

To find the coordinates of point R that divides the line segment AB in the ratio of 3:1, we can use the section formula. The coordinates of points A and B are given as A(-4, 0) and B(0, 6).

### Step-by-step Solution:

1. **Identify the coordinates of points A and B:**
- A = (-4, 0)
- B = (0, 6)

2. **Determine the ratio in which point R divides the line segment AB:**
- Since R = 3/4 AB, it divides AB in the ratio of 3:1.

3. **Use the section formula:**
The section formula states that if a point R divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m:n, then the coordinates of R are given by:
\[ R\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
Here, m = 3, n = 1, \(x_1 = -4\), \(y_1 = 0\), \(x_2 = 0\), \(y_2 = 6\).

4. **Calculate the x-coordinate of R:**
\[ x_R = \frac{3 \cdot 0 + 1 \cdot (-4)}{3 + 1} = \frac{0 - 4}{4} = \frac{-4}{4} = -1 \]

5. **Calculate the y-coordinate of R:**
\[ y_R = \frac{3 \cdot 6 + 1 \cdot 0}{3 + 1} = \frac{18 + 0}{4} = \frac{18}{4} = \frac{9}{2} \]

6. **Combine the coordinates to find R:**
Thus, the coordinates of point R are:
\[ R(-1, \frac{9}{2}) \]

### Final Answer:
The coordinates of R are \((-1, \frac{9}{2})\).